B: Match the objects which are similar in shape. Ans:
C: Tick ✅ the shape which is similar to ball
Ans: D: Tick ✅ the shape which is similar to cap
Ans:
E: Tick ✅ the shape which is similar to glass Ans:
F: Tick ✅ the shape which is similar to matchbox. Ans:
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Let us Do
Page No 13
A: Make a house, toy, tower, robot, bus or anything you like using different objects in your surroundings. You can also use notebooks, books, pencil box, water bottle, waste or old boxes, birthday caps, funnels, etc. Ans:
Also read: 3-Days Study Plan: What is Long? What is Round? (Shapes)
Let us Slide
Page No. 16
A: Write ‘R’ for rolling objects and ‘S’ for sliding objects in the given in the below picture.
Ans:
B: Collect different objects from your surroundings and see if they roll or slide. Ans:
C: Do you see things which can do both, roll and slide? If yes, discuss in the class. Ans:
Chapter Notes: Our Home: Earth, a Unique Life-Sustaining Planet
What would Earth be like if it had never known life at all?
Our planet is not just another ball of rock in space. It is the only known place in the universe that supports a vast variety of life — from towering mountains to deep oceans and green forests. With the help of Earth observation satellites such as those operated by ISRO, scientists study the special features that make Earth a life‑sustaining planet.
Image by Earth Observation Satellite (ISRO)
In this chapter you will explore the conditions and systems that make Earth uniquely fit for life. You will review Earth’s physical structure, its place in the Solar System, how living and non‑living things interact, and the challenges that threaten life on Earth.
About the Image:
The image was captured by an ISRO Earth Observation Satellite.
It is a mosaic made by combining nearly 3,000 smaller images.
This is a false‑colour image — special colours are used to highlight specific information.
Such images help scientists study land, water, plant growth, and environmental changes more clearly.
Why Is Earth a Unique Planet?
Out of the billions of planets in the universe, Earth is the only one we know that has life in so many forms — plants, animals, people, and microorganisms.
Earth’s Crust is like the thin skin of an apple
All living things exist on a very thin layer on Earth’s surface called the crust.
If Earth were the size of an apple, the crust where life exists would be as thin as the apple’s skin.
Below the crust there are other layers: upper mantle, lower mantle, outer core, and inner core, but life is restricted mostly to the crust and the near‑surface parts of the mantle and atmosphere.
Activity 13.1: What Makes Earth Special?
Think about and list interesting features of Earth that matter for our lives. For example:
Here is a completed table you can discuss and add to:
Although the crust is thin, it provides everything needed for life: air, water, soil, and resources.
Earth supplies air to breathe, water for drinking and agriculture, and soil for growing food.
Materials such as rocks, timber, and metals are available for building homes, roads and tools.
Planets of Our Solar System
The Solar System has eight planets that orbit the Sun in nearly circular paths.
Jupiter, Saturn, Uranus, Neptune: Larger, mostly made of gases and ices (gas giants and ice giants).
Activity 13.2: Comparing Planets
Collect and compare information about:
The average temperature of each planet.
Relative size (radius) compared to Earth.
Whether it has an atmosphere.
Observations:
All planets receive energy from the Sun.
Planets closer to the Sun are generally hotter; those further away are colder.
Why Is Venus the Hottest Planet?
Venus is not the closest planet to the Sun, but it is the hottest.
Its atmosphere is very thick and is made mainly of carbon dioxide (CO2).
Carbon dioxide traps heat through the greenhouse effect, so Venus retains large amounts of heat.
On Earth, the greenhouse effect also warms the surface, but it is much weaker and essential for maintaining temperatures suitable for life.
Difference between planetary greenhouse and a plant greenhouse:
Planetary greenhouse: Gases in the atmosphere trap infrared radiation (heat) emitted from the surface.
Plant greenhouse: Glass walls reduce air circulation and trap warm air physically.
Both keep places warm, but the mechanisms differ.
What Makes the Earth Suitable for Life to Exist?
Several physical and chemical properties of Earth combine to make it habitable. These include its position in the Solar System, size and atmosphere, and its magnetic field.
1. Position of Earth — The
Habitable Zone
Earth’s distance from the Sun is just right — neither too close nor too far.
This distance keeps temperatures such that water mostly exists as a liquid, which is essential for all known life.
If Earth were closer to the Sun, it would be too hot and water would evaporate.
If Earth were further from the Sun, it would be too cold and water would freeze.
The region around a star where liquid water can exist is called the habitable zone or the Goldilocks zone.Habitable Zone
Over 70% of Earth’s surface is covered with water, giving Earth the name the Blue Planet.
Blue Planet
Mars and the Possibility of Life
Mars
Mars lies near the edge of the Sun’s habitable zone.
Many spacecraft and rovers have studied Mars; no proof of current life has been found yet.
Evidence suggests Mars may once have had liquid water and possibly simple life. Future missions may provide new information.
2. Size of Earth and Its Atmosphere
Earth’s nearly circular orbit helps keep sunlight and temperatures fairly steady throughout the year.
If Earth were smaller, its gravity might be too weak to hold a dense atmosphere; gases could escape to space (as on Mercury and Mars).
If Earth were much larger, very strong gravity might make conditions unsuitable for life as we know it.
The atmosphere supplies oxygen for breathing and carbon dioxide for plants.
Some oxygen in the upper atmosphere forms ozone, creating the ozone layer that shields life from harmful ultraviolet (UV) radiation.
The atmosphere also contributes to the greenhouse effect, which traps enough heat to keep Earth warm but not too hot.
Our Scientific Heritage: Exploring Mars
Mangalyaan
Mangalyaan (Mars Orbiter Mission), launched by ISRO in 2013, is an important Indian mission to study Mars.
The mission studied Mars’ atmosphere and surface and searched for signs of past water and conditions that could have supported life.
Mangalyaan demonstrated how effective, low‑cost space missions can deliver valuable scientific results.
3. Magnetic Field of the Earth
Along with Earth’s right position, size, and atmosphere, the magnetic field is another key factor that helps make Earth safe for life.
Earth behaves like a giant magnet; a freely suspended magnet (a compass needle) points toward the magnetic north and south.
The region around a magnet where its influence is felt is called the magnetic field.
Scientists believe the motion of molten iron in the Earth’s outer core generates the magnetic field.
Why Earth’s Magnetic Field Is Important
Earth is constantly hit by tiny, high‑energy particles from space:
Cosmic rays from distant space.
Solar wind, composed of charged particles emitted by the Sun.
These particles can damage the atmosphere, reduce the ozone layer, increase harmful UV radiation and harm living organisms.
The magnetic field acts as a shield by deflecting many of these charged particles away from Earth.
This protection helps preserve the atmosphere and makes Earth safer for life.
What Allows Life to Be Sustained on Earth?
It is the connections between living (biotic) and non‑living (abiotic) parts of Earth that enable life to thrive.
Air, Water and Sunlight
The atmosphere provides oxygen for respiration and contains carbon dioxide used by plants for photosynthesis.
Sunlight supplies energy for photosynthesis and warms Earth’s surface. Part of this heat is trapped by the atmosphere (greenhouse effect), keeping temperatures suitable for liquid water.
Water covers nearly 70% of Earth and forms the hydrosphere(oceans, seas, rivers, lakes, groundwater). Water:
dissolves and transports nutrients,
helps animals regulate body temperature and digestion,
is essential for photosynthesis and all cellular processes.
Freshwater is required for agriculture and human consumption.
Water vapour forms clouds that produce rain and snow, which refill rivers, lakes, and groundwater.
Air movement (wind) shapes weather and rainfall patterns; this influences farming and water supply.
Soil, Rocks and Minerals
The geosphere (Earth’s crust) contains rocks, soil and minerals that support life.
Soil provides a medium for plant growth and contains nutrients such as nitrogen and potassium, released by weathering of rocks and decomposition of dead organisms.
Minerals in rocks and soil are resources for salt, coal, oil, iron, copper and many other materials essential for human life and technology.
Geodiversity — a variety of landforms, rocks and soils — creates different habitats and supports biodiversity.
Non‑living parts of nature actively shape ecosystems and biological communities.
Plants, Animals and Microorganisms
The biosphere is the zone where life exists — including land, water and the lower atmosphere. It contains all living organisms: trees, grasses, herbs, animals, insects and microscopic organisms such as bacteria and fungi.
Plants perform photosynthesis, using sunlight, water and carbon dioxide to make food and release oxygen.
Animals depend on plants and other animals for food and energy.
Microorganisms (decomposers) break down dead matter and recycle nutrients into the soil for use by plants.
These interactions form complex food chains and food webs that maintain ecological balance — the essence of ecology.
The Importance of Balance
Earth is an interconnected system: land, air, water and living organisms interact constantly.
Small changes — such as deforestation — can alter rainfall, soil quality, air composition and wildlife populations.
Maintaining balance in these systems is essential for a healthy, habitable planet.
If organisms did not reproduce, species would eventually disappear. Reproduction ensures continuity of life across generations.
What is Reproduction?
Reproduction is the biological process by which organisms produce new individuals of the same kind.
Parents pass down instructions called genes (genetic material) that determine the form and functioning of an organism.
Genes are found in cells and guide the development of body parts and biological processes.
Why is Reproduction Important?
It keeps each species going generation after generation.
It allows variation — small changes in genes — that can help organisms adapt to changing environments.
Accumulation of such changes over many generations can lead to evolution of new traits or species.
Examples: Camels developed humps to survive desert conditions; microbes can evolve antibiotic resistance.
How Can Offspring Be Similar Yet Different?
Reproduction can produce offspring that resemble their parents yet display differences.
There are two main types of reproduction:
Asexual reproduction: New individuals arise from a single parent and are genetically almost identical to it.
Sexual reproduction: Two parents contribute genetic material, producing offspring with mixed traits from both parents.
Asexual Reproduction
In asexual reproduction, one parent produces new individuals that are genetic copies of the parent.
Example: Many plants can reproduce by vegetative propagation — planting a part of the plant (leaf, stem, root) and growing a new plant.
Activity 13.3: Vegetative Propagation in Plants
Plant stem cuttings (e.g. money plant), potato eyes, or pieces of ginger in moist soil.
Provide water, air and sunlight.
Observe daily when roots, shoots and new leaves appear.
This demonstrates how some plants grow new individuals from a part of the original plant.
Other examples of asexual reproduction:
Bacteria, amoeba: Binary fission — one cell divides into two identical cells.
Hydra: Budding — a new individual grows from the body and detaches.
Planaria, some algae: Regeneration — they can regrow from fragments.
Sexual Reproduction
Sexual reproduction is the process in which two parents (usually male and female) produce offspring.
This is common in most animals and flowering plants. Some microorganisms also have mating types that act like parents.
Special Cells for Reproduction: Gametes
Both parents produce special reproductive cells called gametes.
Male gametes: sperm in animals; pollen in flowering plants.
Female gametes: egg in animals; ovule in plants.
Gametes carry half the genetic information of each parent. When a male and female gamete fuse (fertilisation), they form a zygote with a full set of genes.
Why Don’t Babies Look Exactly Like Their Parents?
Offspring receive a unique mix of genes from both parents.
That is why they resemble their parents but are not exact copies.
Siblings may look different because each receives a different combination of genes.
Mixing of genes increases variation, which is useful for adaptation and evolution.
Ovule (female), inside the ovary, contains the female gamete.
Pollination: Transfer of pollen from anther to stigma, often by wind, insects or animals.
Fertilisation: Pollen reaches the ovule and male and female gametes fuse to form a zygote.
The zygote develops into a seed.
The ovule becomes the seed and the ovary develops into fruit.
Seed dispersal: Fruits or seeds are carried away by animals, wind or water. Seeds that fall in suitable places germinate using stored food to grow roots and shoots.
Sexual Reproduction in Animals
Animals have two types of reproductive cells: sperm (male) and egg (female).
Fertilisation occurs when sperm and egg unite to form a zygote.
In fish and amphibians (e.g. frogs), fertilisation often takes place externally in water, with parents releasing eggs and sperm into water.
In birds and most mammals, fertilisation occurs inside the female’s body. Sperm swim to meet the egg.
After fertilisation:
Birds lay eggs; the embryo develops using food stored in the egg until hatching.
Mammals (most) give birth to live young; the embryo develops inside the mother who supplies food and oxygen.
Main difference:
Egg‑laying animals provide food for the embryo inside the egg.
Mammals provide food to the embryo inside the mother’s body.
What Are the Threats to Life on Earth?
Earth’s life depends on a delicate balance between living things (plants, animals, microbes) and non‑living things (air, water, soil, sunlight). Human activities are disturbing this balance, leading to major environmental problems.
The three main global challenges today are:
Climate change
Biodiversity loss
Pollution
1. Climate Change
Burning fossil fuels (coal, oil, gas) releases greenhouse gases like carbon dioxide (CO2) and methane.
These gases trap more heat in the atmosphere, causing global warming.
Normally, CO2 is absorbed by plants, trees and plankton in oceans, but excess CO2 from burning fossil fuels adds more heat than the Earth can absorb quickly.
Even a small temperature rise can:
melt ice caps and raise sea levels, causing coastal flooding,
increase extreme weather events such as heavy rainfall, storms, droughts and heat waves,
cause extinction of plants and animals that cannot adapt quickly.
Long‑term changes in temperature, rainfall and weather patterns are collectively known as climate change.
2. Biodiversity Loss
Destroying habitats (forests, grasslands, wetlands) causes plants and animals to vanish.
This upsets food chains and ecosystems:
If grasses disappear, herbivores lose food.
Without herbivores, predators cannot survive.
Each species plays a role; losing species weakens nature’s capacity to support life.
3. Pollution
Air pollution
Comes from factories, vehicles and burning fuels.
Harms human health (respiratory illness), damages crops and causes smog and acid rain.
Water and soil pollution
Caused by industrial effluents, agricultural chemicals and plastic waste.
Harms aquatic life, makes water unsafe, and reduces crop yields.
Polluted soil can introduce toxins into the food chain.
All these problems affect people, animals, plants and ecosystems.
The Importance of Global and Local Action
Small changes in global temperature, atmospheric composition or ozone can endanger life.
Earth’s systems — hydrosphere (water), biosphere (living things), atmosphere (air) and geosphere (rocks and soil) — are connected; harm to one affects the others.
International Agreements and Efforts
Countries have agreed on treaties to protect Earth:
Montreal Protocol (1987): Reduced chemicals (CFCs) that damaged the ozone layer; helped ozone recovery.
Earth Summit (1992): A global meeting that promoted cooperation on environment and development.
Kyoto Protocol (entered into force 2005): Set binding emission reduction targets for some countries.
Paris Agreement (2015): Countries committed to limit global warming; the goal is to keep warming well below 1.5°C above pre‑industrial levels.
As of 2025, the world has not yet achieved the 1.5°C goal — stronger action is still needed.
How Can We Help? — Individual and Local Actions
Cut down on pollution: Avoid burning waste; reduce vehicle emissions.
Switch to cleaner energy: Use solar, wind and other renewable sources instead of coal and oil.
Use energy and water carefully: Turn off lights, repair leaks, and use public transport or cycle where possible.
Reduce, reuse and recycle: Repair items instead of throwing away; recycle paper, plastic, glass and metal.
Practice sustainable farming and waste management: Use organic methods, reduce chemical use and manage sewage properly.
Protect biodiversity: Conserve habitats, plant native trees and support protected areas.
Community action: Local communities managing natural resources wisely can make a large positive difference.
Final Summary
Earth is uniquely suited for life because of its right distance from the Sun, suitable size and gravity, atmosphere and magnetic field.
Life depends on continuous interactions among air, water, soil, rocks and living organisms.
Reproduction — both asexual and sexual — ensures continuation and variation of life.
Human activities have introduced serious threats: climate change, biodiversity loss and pollution.
Global treaties and local actions together can help protect Earth; everyone can contribute through small, practical steps.
How does nature keep everything in balance—and what happens when that harmony is disturbed?
From forests and rivers to animals, people, and the land itself, every part of nature is interconnected.
Sometimes, when forests are cut down or rainfall changes, animals like elephants lose their homes and food, forcing them to move into farms or villages, leading to new problems. Elephants moving in search of food and shelter
Even small changes in one part of nature can affect everything else around it.
In this chapter, you’ll explore how water, sunlight, plants, animals, and even humans are all linked in a web of relationships. You’ll discover why balance in nature is so important, how living things depend on each other, and how human actions can impact the whole system—for better or worse. Let’s get started!
How Do We Experience and Interpret Our Surroundings?
Different habitats have different kinds of plants and animals
Habitat:
A habitat is the place where an organism lives.
It provides the surroundings and conditions an organism needs to survive.
Habitats can be small (like tree bark) or large (like a pond or forest).
Diversity in Habitats:
Different habitats have different kinds of plants and animals (living beings).
Organisms adapt to survive in their specific habitats.
Activity: Explore two nearby habitats and identify both the living organisms and the non-living components in each.
Select two nearby habitats (e.g., a pond and a forest).
List the living and non-living components in each habitat.
Common Characteristics of Habitats
Both habitats (like a pond and a forest) have:
Living beings (biotic components): plants, animals, and other organisms.
Non-living things (abiotic components): air, water, sunlight, soil, temperature, stones, etc.
Similarities:
Both have biotic and abiotic components.
Differences:
The types of living beings vary (e.g., fish in a pond, trees in a forest).
The types of non-living components also differ (e.g., more water in a pond, more soil and air in a forest).
Biotic and Abiotic Components
Biotic components: All living things in a habitat (plants, animals, microbes).
Abiotic components: All non-living things in a habitat (sunlight, air, water, soil, temperature).
Why do some organisms live on land and others in water?
Every organism needs certain conditions to survive, such as food, water, oxygen, shelter, and space.
Example: Fish live in ponds because they get food, oxygen, and shelter there. Pond water provides both biotic needs (food from plants/animals) and abiotic needs (oxygen from water).
Many other creatures (frogs, turtles, snakes, insects, birds, plants) share the pond. Each interacts with others and with the non-living parts of the habitat.
Coexistence and Harmony in Habitats
Each habitat has its own unique mix of living (biotic) and non-living (abiotic) components, like air, sunlight, water, soil, and temperature.
Different species in the same habitat might use resources differently: For example, in a forest, a snake may be active at night, while a rodent is active during the day—helping both survive in the same place but at different times.
Who All Live Together in Nature?
Population
Definition: A population is a group of the same kind of organisms (same species) living together in a specific habitat at a given time.
Example: All the fish of the same species in a pond form a fish population.
How Do We Measure Population?
To find the population of a certain plant or animal, you:
Mark a fixed area (for example, 1m × 1m in your school garden).
Count the number of each type of organism (plants or animals) present in that area.
Record this information as the population for that type of organism in that particular area and time. Population means the number of individuals of a particular kind (species) living in a defined area at a certain time.
Community
A community is formed by different populations (different types of plants, animals, and microorganisms) living together and interacting in the same habitat.
It includes all living (biotic) components of the habitat.
Habitat
Definition: A habitat is the physical place or environment where an organism lives.
If there is only one type of organism in a habitat, there will be competition for resources like food, water, and space. This can cause shortages and make it hard for other creatures to survive.
Diversity in a habitat (many types of organisms living together) helps maintain a balance and supports survival by providing different roles and interactions.
Pollination
Flower Structure: Flowers have sepals, petals, stamens (male part), and carpels (female part).
Pollination: The process of transferring pollen grains from the stamen of one flower to the carpel of the same or another flower.
How it happens: Wind, water, insects, bats, and birds help carry pollen.
Why it matters: Pollination is essential for fruits and seeds to form.
Try yourself:
What is a habitat?
A.A type of animal
B.A process of transferring pollen
C.A group of organisms of the same species
D.A physical place where an organism lives
Does Every Organism in a Community Matter?
Yes! Every organism in a community plays an important role, helping maintain balance and supporting the survival of other organisms. Here’s how the activities and scientific studies help us understand this:
Explanation Using the Pond Example:
Pond A (with fish): There are fewer dragonflies because fish eat the dragonfly larvae in the pond.
Pond B (without fish): There are more dragonflies since nothing eats their larvae.
What happens next?
Dragonflies are predators of bees, butterflies, and other insects that help pollinate flowers.
Fewer dragonflies in Pond A means more bees and butterflies can survive and pollinate the flowers.
More pollinators = more pollination, which leads to more flowers producing seeds and more plants.
Scientific studies confirm this: researchers found that plants around ponds with fish were better pollinated than those around ponds without fish, mainly due to this chain of effects.
What Does This Show?
Organisms are interconnected. The presence or absence of one type (like fish) can affect many others (like pollinators and plants).
Every member of a community has a role (niche): Fish control dragonfly populations, dragonflies affect pollinators, pollinators are essential for plant reproduction, and plants provide food and shelter for all.
This is known as a food web or ecological balance, where changes in one group lead to effects (sometimes called a “cascade”) throughout the community.
Impact of Overfishing by Humans
Overfishing removes too many fish from ponds or oceans.
If too many fish are taken away, dragonfly numbers might rise because their predators have gone. This could lower the number of pollinators, resulting in less pollination for nearby plants.
Removing key species like fish upsets the delicate balance of the community, affecting not just other animals but also plants and even the non-living (abiotic) environment.
What Are the Different Types of Interactions Among Organisms and their Surroundings?
Organisms do not live alone. They constantly interact with both living (biotic) and non-living (abiotic) components in their environment.
1. Interactions Between Biotic and Abiotic Components
Definition: Interactions between living things and non-living things in their environment.
Examples:
Plants need sunlight (abiotic) for photosynthesis, water and soil for growth, and air for respiration.
Earthworms live in moist soil.
Fish lay eggs in water (abiotic).
Soil provides nutrients for plants.
2. Interactions Among Abiotic Components
Definition: Interactions between non-living things, which affect the conditions of a habitat.
Examples:
Sunlight warms up the day, increasing the temperature.
Water evaporation happens faster in strong sunlight.
Air currents create gentle waves on the water.
3. Interactions Among Biotic Components
Definition: These are the relationships between living organisms in a community.
Examples:
Frogs eat insects (food chain).
Water snakes eat fish.
Frogs and fish compete for larvae.
Many microbes (tiny living beings) interact in the pond, breaking down dead material.
Plants provide food and shelter for animals.
Mushrooms (fungi) decompose dead plants and animals.
The Concept of Ecosystem
An ecosystem is made up of all the living (biotic) and non-living (abiotic) things in a particular area and all the interactions among them.
Aquatic (water-based) ecosystems include ponds, rivers, and lakes
Terrestrial (land-based) ecosystems include forests, grasslands, and farmlands.
Ecosystems can overlap—for example, a river running through a forest.
Types of Consumers and Producers
Producers (Autotrophs): Make their own food (usually plants via photosynthesis).
Consumers (Heterotrophs): Depend on others for food. – Herbivores: Eat only plants (deer, horse). – Carnivores: Eat only animals (vulture, shikra). – Omnivores: Eat both plants and animals (fox, mouse).
Decomposers: Organisms like mushrooms and bacteria that feed on dead plants and animals, recycling nutrients back into the ecosystem.
Who Eats Whom?
Food Chain: A food chain is a simple sequence showing “who eats whom” in an ecosystem.
Example (Grassland Ecosystem): 1. Grass → Hare →TigerGrass is eaten by the hare; hare is eaten by the tiger. 2. Grass → Grasshopper → Frog → Snake → Eagle Here, each organism is eaten by the next one in the chain.
Trophic Levels
Each organism in a food chain occupies a specific position, called a trophic level:
First trophic level: Producers (plants, e.g., grass, millet)
Second trophic level: Herbivores (organisms that eat plants, e.g., hare, mouse)
Third trophic level: Small carnivores (those that eat herbivores, e.g., frogs)
Fourth trophic level (and above): Large carnivores or top predators (e.g., eagle, hawk, fox)
Trophic Level Pyramid
Ecological Pyramid
When you count the number of organisms at each level (e.g., many grasses, fewer mice, only one eagle), and arrange these numbers with the highest at the base and lowest at the top, you get a pyramid shape.
Producers are always the base, and top predators are at the top, indicating energy loss at each step.
Food Web
In reality, feeding relationships are not simple chains—they’re much more complex, forming a food web.
The grass can be eaten by rabbit or mouse.
Grasshoppers can be eaten by the bird or frog.
Owls can eat mice or frogs.
Because each organism may be eaten by two or more types of organisms, the food chains overlap and link together, making a web. Food Web
Try yourself:
What is the main topic of the text?
A.Food chains
B.Weather patterns
C.Animal habitats
D.Plant growth
What Happens to Waste in Nature?
What are decomposers?
Definition: Decomposers are organisms that break down dead plants, animals, and animal waste into simpler substances, returning nutrients to the soil.
Examples: Mushrooms (a type of fungi), bacteria, beetles, and flies. Examples of Decomposers
How Decomposition Works
When plants, animals, or their waste die, decomposers feed on them.
Fungi (like mushrooms) and bacteria break down complex substances in these dead materials into simpler forms.
Beetles and flies often feed on things like animal dung (e.g., elephant dung), breaking it down further.
This entire process is called decomposition.
Importance of Decomposers
Nutrient Recycling: Decomposition returns important nutrients to the soil, which plants use to grow.
Balance in Nature: Decomposers prevent the buildup of dead materials and waste in the environment.
No Waste in Nature: Nothing truly goes to waste in nature—everything is reused in one form or another thanks to decomposers.
Saprotrophs: Another name for decomposers. Sapro means “rotten,” and troph means “food.”
What are migratory birds?
Birds that travel thousands of kilometers between different habitats and countries to avoid harsh climates or find food.
Example: Demoiselle Cranes visit Khichan village in Rajasthan every winter.
Roles of Migratory Birds
Enhance Beauty: Add color and vibrancy to habitats.
Ecosystem Balance: Act as pollinators (helping flowers reproduce) and seed dispersers, linking different habitats.
Pest Control: Feed on insect pests, helping farmers by reducing crop damage.
How Does One Change Lead to Another?
If plants in a pond die (e.g., due to pollution), then:
Less oxygen is produced in the water.
Fish population declines, as they need oxygen.
Fewer fish means more insects (fish usually eat them).
Extra insects spread to nearby farms and harm crops.
Farmers use more pesticides, which further harms the environment.
One small change (like plant death) causes a chain reaction or “cascading effects” through the ecosystem.
Effects of Human Intervention: The Frog Leg Export Story
In the 1980s, India exported a huge number of Indian bullfrogs.
This reduced frog populations.
Fewer frogs meant more insects, including pests in farms.
Farmers then used more pesticides, harming the environment, water, soil, and health of living beings.
The Government banned frog-leg exports to help restore balance.
Ecosystem Balance
Interactions between organisms and their environment keep populations and resources stable—this is called ecological balance.
This balance is always changing (dynamic), but large disruptions (such as overuse, pollution, loss of species) can harm it.
How Do Interactions Maintain Balance in Ecosystems?
Besides feeding relationships, organisms also compete for common resources like food, water, physical space, or sunlight. This competition helps control population size and keeps the ecosystem balanced. Without it, one species could multiply too much causing an imbalance in the ecosystem
Other Relationships:
Mutualism: Both organisms benefit (e.g., bees and flowers).
Commensalism: One benefits, the other is unaffected (e.g., orchids on trees).
Parasitism: One benefits, the other is harmed (e.g., ticks on dogs).
Benefits of an Ecosystem
Forests: Provide clean air, fertile soil, food, timber, medicines, and beauty.
Water bodies: Give water and food.
All ecosystems: Offer aesthetic value, recreation, and support human well-being.
Mangroves (e.g., Sundarbans): Protect against floods and storms, absorb carbon dioxide, support unique wildlife; recognized as a World Heritage Site.
Threats to Ecosystems
Deforestation, pollution, unsustainable land use, and illegal hunting are harming all types of habitats.
The Sundarbans mangroves are under threat from wood cutting, pollution, and resource overuse.
Such activities disrupt natural cycles and reduce biodiversity.
How To Protect Ecosystems
Protected Areas: Such as national parks, biosphere reserves, and sanctuaries conserve habitats and wildlife. Examples: Jim Corbett, Manas, Nilgiri Biosphere Reserve, Chilika Lake.
Community Action: People must work together to conserve resources, avoid pollution, and preserve natural areas.
Human-Made (Artificial) Ecosystems
Examples: Farms, fish ponds, parks.
When managed well, they can help reduce pollution, support biodiversity, and provide recreational spaces.
Need: Continuous care and human management.
Try yourself:
What is the main purpose of interactions?
A.To create chaos
B.To maintain balance
C.To confuse others
D.To avoid communication
What Are the Benefits of an Ecosystem?
Ecosystems consist of biotic (living) and abiotic (non-living) components that depend on each other to support life processes.
Humans benefit from ecosystems in many ways:
Forests provide fresh air, fertile soil, food, fibres, timber, and medicines.
Aquatic ecosystems provide water and food.
Ecosystems also offer aesthetic (beauty) and recreational (enjoyment) value.
This supports human well-being and shows the close connection between nature and humans.
However, overusing or misusing natural resources disturbs the balance in nature.
Real-Life Example: The Sundarbans – A Threatened Ecosystem
The Sundarbans have the largest mangrove forests in the world.
Located where the Ganges and Brahmaputra Rivers meet, between India and Bangladesh.
Home to various flora (plants) and fauna (animals), many of which are endangered.
Protects against storms and floods by slowing down strong winds and waves.
Trees absorb carbon dioxide from the air and release oxygen.
Declared a World Heritage Site by UNESCO (United Nations Educational, Scientific and Cultural Organization) in 1987 due to its importance.
Threats
Mangrove trees are cut for fuelwood and farming.
Illegal hunting and overuse of forest resources threaten wildlife.
Pollution from industrial waste and untreated sewage damages water and habitat.
These human activities disrupt the natural functioning of ecosystems.
Other Threatened Ecosystems in India Ecosystems across India (forests, rivers, scrublands, wetlands, grasslands, coastal areas) are under threat.
Problems:
Deforestation (cutting trees).
Overuse of natural resources.
Spread of invasive species (non-native plants/animals that harm locals).
Unsustainable land use.
Pollution.
Call to Action: Think about actions you and your community can take to protect forests, rivers, and wetlands to stop damaging them.
Protected Areas for Conservation
Definition: Protected areas are parts of land or water set aside to conserve wildlife and their habitats.
India has many protected areas: national parks, wildlife sanctuaries, biosphere reserves, and community conserved areas.
Benefits: They protect entire habitats, including endangered animals, birds, and rare plants.
Famous Examples:
Jim Corbett National Park (Uttarakhand).
Manas National Park (Assam).
Nilgiri Biosphere Reserve (Western Ghats).
Chilika Lake (Odisha).
Eaglenest Wildlife Sanctuary (Arunachal Pradesh).
Hemis National Park (Leh).
Keibul Lamjao National Park (Manipur).
Pirotan Island Marine National Park (Gujarat).
Protected areas play a big role in saving nature for future generations.
Our Scientific Heritage
The ancient text Vrikshayurveda emphasizes soil health and nourishment.
It advocates for continuous soil nourishment through organic manure like Kunapa Jala (a liquid fertilizer made from animal and plant waste by fermentation, which breaks complex substances into simpler ones) and other composted materials.
Human-Made Ecosystems
Humans create artificial ecosystems like fish ponds, farms, and parks to meet their needs.
When well-designed, they reduce pollution, support biodiversity (variety of life), and provide recreational spaces.
Unlike natural ecosystems, these need human care and management.
How Do Healthy Ecosystems Serve Our Farms?
Farming is a major livelihood in India but can become unsustainable without environment-friendly practices.
Humans have practiced farming for thousands of years to grow food.
As population grew, dependence on agriculture increased.
Between 1950 and 1965, India faced a food crisis due to low crop production.
In the mid-20th century, the Green Revolution used tractors, machines, synthetic fertilizers, and pesticides to increase food production.
However, these methods are now seen as unsustainable due to:
Overuse of synthetic chemicals.
Excessive groundwater extraction.
Growing only one type of crop (monoculture) for commercial gain.
Harms to Environment and Health:
Overusing pesticides and monoculture lead to soil degradation (loss of quality).
Reduces soil fertility by decreasing friendly microorganisms and organic matter (humus), which binds soil particles.
Without humus, soil erodes easily.
Reduces natural predators, increasing pest populations
Heavy irrigation and repeated ploughing disturb soil organisms like earthworms and snails, important for ecological balance.
Pests may develop resistance to pesticides, making control harder.
Monoculture reduces crop diversity and affects pollinators (e.g., bees), crucial for food production.
The Moon can sometimes be seen during the daytime. This made Meera curious when she noticed the Moon in the sky on Makar Sankranti. This chapter explains why the Moon changes its appearance and how people have used observations of the sky to measure time.
Kites in the Sky
How Does the Moon’s Appearance Change and Why?
Phases of the Moon
The changing shapes of the Moon’s illuminated (bright) portion, as seen from Earth, are known as the phases of the Moon.
Waning Period (Krishna Paksha)
After a full Moon, the bright portion decreases from a full circle to a half circle in about a week.
The bright portion continues to shrink, disappearing completely in another week.
This two‑week shrinking period is called the waning period or Krishna Paksha in India.
New Moon (Amavasya)
The day when the Moon is not visible at all is called the new Moon or Amavasya.
Waxing Period (Shukla Paksha)
After the new Moon, the bright side grows to a half circle in about a week and becomes a full circle (full Moon) in another week.
This two‑week growing period is called the waxing period or Shukla Paksha in India.
The Moon’s waxing and waning occurs in a cyclical (repeating) pattern each month.
The full cycle from one full Moon to the next takes about one month (about 29.5 days).
Important Terms
Full Moon (Purnima): The day when the Moon appears as a full bright circle.
New Moon (Amavasya): The day when the Moon is not visible at all.
Waxing: The period when the bright part of the Moon increases (from new Moon to full Moon).
Waning: The period when the bright part of the Moon decreases (from full Moon to new Moon).
Gibbous: More than half but not fully illuminated Moon.
Crescent: Less than half illuminated Moon.
Locating the Moon
The Moon’s position in the sky changes each day, even at the same clock time.
On full Moon day:
The Moon is nearly opposite the Sun.
When the Sun rises in the east, the Moon is almost setting in the west.
After full Moon:
Each morning at sunrise, the bright part of the Moon gets smaller and the Moon appears closer to the Sun’s position in the sky.
When the Moon looks like a half circle, it is overhead at sunrise.
A few days after, the crescent Moon appears even closer to the Sun.
The phase (shape) of the Moon and whether it is waxing or waning help you know where and when to look for it in the sky.
Waxing Moon: Best seen at sunset.
Waning Moon: Best seen at sunrise.
The moonrise time becomes about 50 minutes later each day, which explains why the Moon can sometimes be seen during daylight (for example, in the afternoon between 2:00–4:00 p.m.).
After moonrise, wait about 30 minutes for the Moon to climb higher into the sky for an easier view.
Making Sense of Our Observations – The Moon’s Changing Appearance
The Moon’s Shape:
The actual shape of the Moon does not change; what changes is how much of the illuminated part we can see from Earth.
The Moon’s Light:
The Moon does not produce its own light. It appears bright because it reflects sunlight that falls on it.
Sunlit and Dark Halves:
At any moment, half of the Moon faces the Sun and is illuminated by sunlight.
The other half, facing away from the Sun, remains in darkness (non‑illuminated).
Appearance from Earth:
As the Moon revolves around the Earth, the position and viewing angle change.
Although always one half of the Moon is sunlit, that half is not always fully visible from Earth.
The portion of the Moon we see from Earth may be fully illuminated, partly illuminated, or not illuminated at all — producing the phases.
Full Moon and New Moon:
Full Moon: When the entire sunlit half faces Earth, we see the Moon as a whole bright circle.
New Moon: When the non‑illuminated half faces Earth, we cannot see the Moon at all.
Reason for Changing Appearance:
The Moon seems to change shape because its position relative to Earth and Sun is constantly changing as it orbits Earth.
We only see the illuminated part that faces toward us; the visible fraction changes with the Moon’s orbit.
The Moon’s Phases with a Simple Model
Model Demonstration:
Use a small ball on a stick to represent the Moon, a torch or lamp as the Sun, and your head as the Earth.
Hold the ball slightly above your head and shine the torch toward the ball to represent sunlight.
As you turn in a circle, the ball (“Moon”) shows a changing illuminated portion to your eyes, similar to lunar phases.
What the Model Shows:
Full Moon: When the ball is held opposite the lamp (behind you compared to the Sun), the side facing you is fully lit — like a full Moon.
New Moon: When the ball is held between your head and the lamp (towards the Sun), you see only the dark side — like a new Moon.
Crescent and Gibbous Phases: Turning the ball slowly changes the visible portion: sometimes crescent (less than half lit), sometimes gibbous (more than half lit).
The line between the bright and dark parts is always curved — this matches what is seen on the real Moon.
The Science Behind the Phases:
At every moment, half the Moon is lit by sunlight and half is dark.
As the Moon revolves around the Earth, the angle between Earth, Moon and Sun changes, so the part of the Moon we see bright changes accordingly.
Phase Names:
Crescent: Less than half illuminated.
Gibbous: More than half illuminated.
Full Moon: Whole face illuminated.
New Moon: No illuminated part visible.
Why Do Moon Phases Occur?
Incorrect Idea: Moon phases are not caused by Earth’s shadow falling on the Moon.
Correct Reason: The phases of the Moon occur because of the changing relative positions (orientation) of the Sun, Moon and Earth as the Moon revolves around Earth.
Earth’s Shadow and Lunar Eclipse:
The only time Earth’s shadow actually falls on the Moon is during a lunar eclipse.
Lunar eclipses can only happen on a full Moon day.
Solar eclipses can only happen on a new Moon day.
Why Don’t Eclipses Happen Every Month?
Eclipses do not occur every month even though there is a full Moon and new Moon monthly.
This is because the Moon’s orbit is slightly tilted relative to Earth’s orbit around the Sun.
Most months, the Sun, Earth and Moon do not line up perfectly for the Earth’s shadow to cover the Moon (lunar eclipse) or for the Moon’s shadow to fall on Earth (solar eclipse).
How Did Calendars Come into Existence?
Natural Cycles and Time Measurement
The apparent daily motion of the Sun (rising in the east, setting in the west) is due to Earth’s rotation on its axis.
This natural cycle forms the basis of the day — the primary unit of timekeeping.
Mean Solar Day: The time between one “highest Sun position” (shortest shadow at noon) to the next is about 24 hours, known as the mean solar day.
Shadow Tracking and the Day
The shortest shadow during the day marks the Sun’s highest point in the sky (noon).
Measuring from one day’s noon to the next gives the length of a day.
The average solar day is about 24 hours.
The Month and the Moon
The phases of the Moon create another natural cycle — one complete phase cycle (from full Moon to next full Moon) takes about 29.5 days (approximately one month).
This lunar cycle is the basis for measuring a month.
The Year and the Seasons
One year is the time Earth takes to make a full revolution around the Sun, which is about 365¼ days.
The repetition of seasons (spring, summer, autumn, winter) marks the annual cycle.
Try yourself:
What is the primary unit of timekeeping based on Earth’s rotation?
A.Month
B.Year
C.Day
D.Hour
Types of Calendars
1. Lunar Calendars
Based on the phases of the Moon.
Each lunar month is about 29.5 days and 12 lunar months make a year of about 354 days.
This lunar year is shorter than a solar year, so the months shift with respect to the seasons over time.
2. Solar Calendars
Based on Earth’s revolution around the Sun and the arrival of seasons.
The Gregorian calendar is a solar calendar (widely used) with months adjusted so a regular year has 365 days.
To correct the extra quarter day, a leap day is added every 4 years (February 29).
Leap year rule: Years divisible by 4 are leap years, with additional adjustments: skip leap years every 100 years, but add them back every 400 years.
3. Luni‑Solar Calendars
Combine lunar months with corrections to keep in sync with the solar year and seasons.
In India and elsewhere, an extra month (Adhika Maasa) is added every 2–3 years to adjust the calendar with the solar year.
The names of months in Indian luni‑solar calendars include: Chaitra, Vaisakha, Jyeshtha, Ashadha, Shravana, Bhadrapada, Ashwin, Kartika, Margashirsha (Agrahayan), Pausha, Magha, and Phalguna.
Amant calendars: A month begins after a new Moon and ends on the next new Moon.
Purnimant calendars: A month begins after a full Moon and ends at the next full Moon.
Observations and Heritage
Ancient observers noticed 12 cycles of Moon phases fit in one yearly cycle of seasons.
The Sun’s position at sunrise changes through the year (northward in summer, southward in winter) due to Earth’s tilt.
This movement is described in Indian tradition as Uttarayana (northward, roughly December–June) and Dakshinayana (southward, June–December).
Solstices and equinoxes are important points for tracking the Sun’s yearly journey and for adjusting calendars.
The Indian National Calendar
The Indian National Calendar, also known as the Saka Calendar, is a solar calendar officially used by the Government of India along with the Gregorian calendar.
It consists of 365 days in a year in a normal year.
The year begins on 22 March (the day after the spring equinox). In leap years it begins on 21 March.
Each month is named after traditional Indian months: Chaitra, Vaisakha, Jyestha, Ashadha, Shravana, Bhadrapada, Ashwin, Kartika, Agrahayana, Pausha, Magha, and Phalguna.
Months have 30 or 31 days: In a regular year, months 2–6 have 31 days; the rest have 30 days.
In a leap year, a day is added to Chaitra (the first month), making it start on 21 March.
The Indian National Calendar was introduced in 1956, based on recommendations by the Calendar Reform Committee (CRC) headed by the astrophysicist Meghnad Saha.
The calendar follows principles that relate to ancient Indian astronomical works such as the Surya Siddhanta.
This calendar helps unify civil timekeeping across India.
Are Festivals Related to Astronomical Phenomena?
Many Indian festivals are linked to the phases of the Moon and thus are based on lunar or luni‑solar calendars.
Examples
Diwali: Celebrated on the new Moon of Kartika.
Holi: Celebrated on the full Moon of Phalguna.
Buddha Purnima: Occurs on the full Moon of Vaisakha.
Eid‑ul‑Fitr: Celebrated after sighting the crescent Moon at the end of Ramadan.
Dussehra: Occurs on the tenth day of Ashwina.
These festivals appear on different dates in the Gregorian calendar each year because the lunar year and solar year are not the same length.
Luni‑solar calendars add an extra (intercalary) month every few years to align lunar months with the solar year, causing date shifts of less than a month relative to the Gregorian calendar.
Pure lunar calendars do not make this adjustment; so festivals like Eid‑ul‑Fitr move through the Gregorian calendar months over the years.
Solar Festivals
Some Indian festivals follow a solar sidereal calendar, occurring nearly on the same Gregorian date each year.
Examples: Makar Sankranti, Pongal, Bihu, Vaisakhi, Poila Baisakh, and Puthandu.
These were originally linked to solstices or equinoxes, but a slight difference between sidereal and tropical years causes slow drift of dates over centuries (for example, Makar Sankranti shifts by one day about every 71 years).
Variations in Festival Dates
The exact lunar phase at sunrise can vary for eastern and western parts of India, causing festival dates to differ by a day in different regions even in the same year.
To standardise dates, the Rashtriya Panchang (national almanac) is published by the Positional Astronomy Centre, Government of India, giving advanced calculations for official festival dates.
Cultural Connection
The Moon and Sun inspire Indian classical art forms:
Music: Ragas such as Chandrakauns and Chandranandan are inspired by the Moon.
Dance: Gestures (mudras) such as Chandrakala and Ardhachandran in Bharatanatyam and other dances invoke lunar imagery.
Visual Arts: Traditional painting styles (e.g., Madhubani, Warli), sculpture and pottery frequently depict the Moon and Sun, reflecting their cultural significance.
Why Do We Launch Artificial Satellites in Space?
Natural vs Artificial Satellites
The Moon is Earth’s natural satellite, revolving around our planet.
Artificial satellites are man‑made objects placed in orbit around Earth by various countries and organisations.
Artificial Satellites: Purpose and Functions
Appearance: Artificial satellites can appear as small, bright, continuously moving dots across the sky.
Typical Low Earth Orbit: Many satellites orbit at heights of a few hundred kilometres (for example around 800 km) and complete an orbit in roughly 100 minutes.
Scientific research (studying space, atmosphere, and Earth)
Try yourself:
What is one reason we launch artificial satellites into space?
A.To communicate with animals
B.To study weather
C.To explore oceans
D.To grow plants
Satellites and Missions by ISRO (Indian Space Research Organisation)
Cartosat Series: High‑resolution imaging satellites for mapping, city planning, and disaster response. Platforms such as Bhuvan use Cartosat images to analyse soil, land use, vegetation, and terrain.
AstroSat: Observatory satellite for studying stars and other celestial objects.
Chandrayaan 1, 2, 3: Moon missions for exploration and scientific study.
Aditya‑L1: Satellite for studying the Sun.
Mangalyaan: Mars Orbiter Mission.
Student Satellites: ISRO encourages students to build and launch small satellites, such as AzaadiSat, InspireSat‑1, and Jugnu.
Observing Artificial Satellites
How to Spot: Look for a small, bright, continuously moving dot in the sky, typically visible just after sunset or before sunrise.
Satellites can often be seen without a telescope.
Mobile apps and websites for satellite tracking show which satellites are visible at your location and time.
Space Debris (Space Junk)
When artificial satellites and rocket parts become old and stop working, they turn into space debris.
Risks: Collisions with functional satellites and cluttering of useful orbits.
Disposal and Mitigation
Small debris usually burns up while entering the atmosphere.
Larger fragments can survive re‑entry and fall to Earth.
Countries and agencies collaborate on solutions to minimise and remove space debris (for example, controlled re‑entry, moving satellites to graveyard orbits, and debris‑removal research).
Key Figures in the Indian Space Programme
Vikram Sarabhai: Pioneer of India’s space programme, often called the “Father of the Indian Space Programme”.
The Vikram Sarabhai Space Centre (VSSC) in Thiruvananthapuram is named in his honour.
VSSC focuses on rocket and launch vehicle technology.
Key Points to Remember
Phases of the Moon: The changing shapes of the illuminated part of the Moon observed from day to day (new Moon, crescent, half, gibbous, full Moon).
Cause of Moon Phases: Phases occur because we see varying portions of the Moon’s sunlit side as it orbits Earth.
Cycle of Moon Phases: A complete sequence of the Moon’s phases takes about a month (≈29.5 days).
Calendars: Systems created using natural cycles — day (Earth’s rotation), month (Moon’s phases), and year (Earth’s revolution around the Sun).
Lunar Calendar: Follows the cycle of the Moon’s phases and has about 354 days in 12 months.
Solar Calendar: Follows the cycle of seasons determined by Earth’s orbit; the Gregorian calendar is an example.
Luni‑solar Calendar: Adapts lunar months to the solar year by adding an extra month occasionally (as in many Indian traditional calendars).
Artificial Satellites: Human‑made objects launched into orbit to provide communication, navigation, weather and scientific data.
Have you ever wondered why the warning “Objects in mirror are closer than they appear” is written on the side-view mirrors of cars? Or why reading glasses sometimes have a curved line on their lenses? Let’s explore these questions with Meena! On a sunny afternoon during her summer holidays, Meena visited a science centre. Among all the amazing displays, something unusual caught her eye—a row of curved mirrors. MirrorsTo her surprise, when she looked into one, her face seemed comically large, while her brother, just a little farther away, looked upside down! At another mirror, she saw a tiny version of herself staring back.
Why do mirrors behave this way? Through the world of spherical mirrors and lenses, get ready to discover the secrets of light and learn how mirrors can make images appear bigger, smaller, or even flipped around!
What Are Spherical Mirrors?
Spoon as a Mirror—A Simple Observation
A shiny metallic spoon can act like a mirror. You can see your face in it if you hold it close.
If you look at the inner (curved inward) side of the spoon, your image appears inverted (upside down).
If you look at the outer (bulging outward) side, your image appears erect (upright) but much smaller than your real face.
This difference occurs because each side of the spoon curves in a different way, mimicking different types of curved mirrors.
Curved Mirrors and Spherical Mirrors
Mirrors like your spoon can be specially made as curved mirrors for scientific and everyday use.
The most common type of curved mirror is the spherical mirror.
Definition of Spherical Mirrors: Spherical mirrors are mirrors whose reflecting surfaces are shaped like a part of a hollow glass sphere. Reflecting surfaces of spherical mirrors can curve either inwards or outwards.
Types of Spherical Mirrors
1. Concave Mirror
A mirror whose reflecting surface is curved inwards, like the inside of a spoon or a bowl..
Edge bulges out, center dips in — think “cave.”
The outline of this mirror is part of a circle when viewed from the front.
2. Convex Mirror
Reflecting surface curves outwards (like the outer side of a spoon, or the back of a bowl).
Center bulges out, edge curves back.
How Spherical Mirrors Are Made?
Shape: Spherical mirrors have a shape as if they are parts of an imaginary hollow sphere.
Manufacture: Despite their shape, these mirrors are not made by slicing a real hollow glass sphere.
Process: They are actually made by grinding and polishing a flat piece of glass into a curved surface. Then, a reflective coating (like a thin layer of aluminum) is added.
Placement of Reflective Coating:
If the coating is placed on the outer curved surface, the result is a concave mirror.
If the coating is placed on the inner curved surface, the result is a convex mirror.
What Are the Characteristics of Images Formed by Spherical Mirrors?
Concave Mirror:
When object is close (small distance): Image is erect but larger (enlarged) than the object.
When object moves farther: Image becomes inverted. It starts enlarged but gets smaller as distance increases.
Convex Mirror:
At any distance: Image is always erect and smaller (diminished) than the object.
As object moves farther: Image size decreases slightly.
Common in Both: Lateral inversion (left-right reversal) is observed in the images.
Comparison to Plane Mirrors:
Spherical mirrors differ from plane mirrors.
Plane mirrors always form an erect image of the same size as the object.
In concave and convex mirrors, image size changes with object distance.
In concave mirrors, images also invert when the object is moved away.
Distinguishing Spherical Mirrors
Idea: Identify if a mirror is plane, concave, or convex by observing object images.
Concave: Enlarged erect image close up, inverted when far.
Convex: Always erect and diminished.
Plane: Always erect and same size.
Real-Life Uses of Spherical Mirrors
Concave and convex mirrors are used in everyday surroundings.
Concave Mirrors:
Reflectors in torches, car headlights, and scooters (concave shape).
Dental mirrors used by dentists: Provide enlarged view of teeth when held close inside the mouth.
Convex Mirrors:
Side-view mirrors on vehicles: Form erect, smaller images of traffic behind; curved outward for wider road view.
Installed at road intersections or sharp bends: Provide visibility from both sides to prevent collisions.
Used in big stores: Monitor large areas to deter thefts.
Telescopes
Most modern telescopes are reflecting telescopes using curved mirrors.
The main mirror is a large concave mirror.
What Are the Laws of Reflection?
Reflection is the bouncing back of light from a surface, like a mirror. The laws of reflection explain how light behaves when it strikes any mirror—plane (flat), concave (curved inward), or convex (curved outward).
The Two Laws of Reflection
1. First Law of Reflection: Angle of Incidence Equals Angle of Reflection
Definition: The angle at which the incoming light ray hits the mirror (angle of incidence, i) is equal to the angle at which it bounces off (angle of reflection, r). In symbols: i = r.
First Law of Reflection
Key Concepts from Setup:
Use a plane mirror with stand, torch, comb (with black paper to make a thin slit), paper clip for holding, white paper sheet, and black paper strip.
Spread white paper on a table, place mirror upright, and shine a thin beam through the slit onto the mirror.
Adjust the beam to hit at different angles; the reflected beam shifts accordingly.
Terms to Remember:
Incident Ray: The incoming light ray that strikes the mirror.
Reflected Ray: The outgoing light ray that bounces back from the mirror.
Normal: An imaginary line drawn at 90° (right angle) to the mirror at the point of incidence.
Angle of Incidence (i): The angle between the incident ray and the normal.
Angle of Reflection (r): The angle between the reflected ray and the normal.
Light is represented as straight rays (lines with arrows) because light travels in straight lines.
How to Prove First Law of Reflection (Observation Process):
Draw the mirror line, incident ray, reflected ray, and normal at point O.
Measure i and r for different incident angles; record in Table.
Special Case: If the incident ray is along the normal, both i and r = 0 (light bounces straight back).
Inference (First Law): Measurements show i always equals r, no matter the incoming angle—this is the first law of reflection.
Example: Shine a torch beam on a plane mirror at various angles—the bounce angle matches the incoming angle exactly.
2. Second Law of Reflection: All in the Same Plane
Definition: The incident ray, the normal to the mirror at the point of incidence, and the reflected ray, all lie in the same plane.
Key Concepts from Setup:
Use the same materials as in activity above but add a stiff chart paper sheet extending beyond the table edge.
Shine a beam on the mirror; see the reflected beam on the extended flat paper (Figure (a)).
Bend the extended part down along the table edge; the reflected beam disappears (Figure (b)).
Flatten the paper; the beam reappears.
Inference (Second Law): Bending creates a new plane, breaking alignment—the law ensures the rays stay “flat” together for predictable reflection.
Step Further: Even if incident rays come from different directions but hit the same point, the normal remains the same, and all (incident ray, normal, reflected ray) stay in one plane.
Example: On flat paper, you see the full path; bending hides it because the plane changes.
How Laws Apply to Spherical Mirrors
The Laws Are Universal: Both laws (i = r and same plane) apply to all mirrors, including spherical ones.
Key Concepts from Setup:
Use plane, concave, and convex mirrors with stands, torch, comb (multiple slits uncovered for parallel beams, and paper clip.
Shine parallel beams on each mirror one by one.
Observations:
Plane Mirror: Reflected beams stay parallel
Concave Mirror: Reflected beams come together (converge)
Convex Mirror: Reflected beams spread out (diverge).
Inference: Each ray obeys the laws, but the mirror’s curve causes parallel rays to converge (concave) or diverge (convex)—this explains focusing or widening effects.
Concentrating Light with Concave Mirrors
Never look at the Sun or into the mirror reflecting sunlight—it can damage eyes. Focus light only on paper, not on faces or people.
Key Concepts from Setup:
Use a concave mirror and thin paper (e.g., newspaper).
Hold the mirror facing the Sun; direct reflected light onto the paper.
Adjust paper distance for a sharp bright spot.
Keep steady for a few minutes.
Observation: The paper starts burning and produces smoke.
Inference: Concave mirrors converge sunlight to a small point, creating intense heat that can ignite paper—this shows the power of focused reflection.
Step Further (Solar Concentrators): Devices using mirrors/lenses to focus sunlight for heating liquids, making steam for electricity, large-scale cooking, or solar furnaces (even melting steel). Recall electric furnaces from an earlier chapter.
What Is a Lens?
Imagine looking through a flat transparent glass window pane—all objects look the same size and shape. But if the surface of the transparent material is curved, objects may not look the same.
How a Water Drop Acts Like a Lens
Materials: A flat strip of glass or clear plastic (e.g., flat scale), few drops of oil, dropper, water, and a paper or book with printed text.
Key Concepts from Setup:
Spread a few drops of oil (or wax) on the glass/plastic strip and rub to make a thin coating (helps water form a round drop).
Use a dropper or finger to place a small water drop on the oiled/waxed spot.
Observations as Concepts:
The water drop’s surface is curved outward (not flat or curved inward).
Place printed text under the strip so it’s directly below the drop.
Look down through the drop: Letters below appear different—often larger (enlarged) than nearby letters.
Inference: The curved surface of the water drop changes the text’s size, acting like a simple lens.
Definition and Types of Lenses
A magnifying glass is a lens that enlarges small print, making letters look bigger.
Lens Definition: A piece of transparent material (usually glass or plastic) with curved surfaces.
Lenses can be convex or concave, like mirrors.
Convex Lens: Thicker at the middle than at the edges.
Concave Lens: Thicker at the edges than at the middle.
Unlike mirrors, lenses allow light to pass through; we see things through a lens, not reflected in it.
How Objects Look Through Lenses
Materials: A convex lens, a concave lens, a lens holder, and a small object.
Setup:
Place the lens upright in the holder.
Put the object behind the lens (raise it to lens level if needed).
Look through the lens from the other side.
Move the object farther and observe changes; repeat for both lenses.
Observations :
Convex Lens:
At small distance: Object appears erect and enlarged (larger).
As distance increases: Object appears inverted; starts enlarged but gets smaller (diminishes).
Concave Lens:
At any distance: Object always appears erect and diminished (smaller).
Size changes (gets even smaller) as distance increases.
Inference: Distance from the lens affects image size and orientation. Convex lenses can enlarge and invert; concave always diminish and keep erect.
Do Lenses Converge or Diverge Light?
Materials: A thin transparent glass plate, a convex lens, a concave lens, a torch and comb (for multiple parallel beams), paper clip to hold comb, two identical books, and white paper sheets.
Setup:
Use books to hold the glass plate or lens upright between them .
Spread paper on both books.
Shine multiple parallel beams on the glass plate, convex lens, and concave lens one by one
Observations:
Thin glass plate: Parallel beams pass through unchanged.
Convex lens: Beams come together (converge).
Concave lens: Beams spread out (diverge).
Inference: Convex lenses are converging lenses (focus light); concave are diverging lenses (spread light). (Diagrams show rays passing through each.)
Drawing Light Through Lenses
Drawings of Activity show rays passing through: unchanged in glass plate, converging in convex lens, diverging in concave lens.
Can a Convex Lens Burn Paper?
Setup: Use a convex lens instead of a concave mirror in the path of sunrays.
Observation: Yes, you can burn the paper—the lens converges sunlight to a hot point, like a concave mirror.
Inference: Convex lenses focus light to create heat, similar to converging mirrors.
Real-Life Uses of Lenses
Lenses are important and used everywhere.
Eyeglasses: Help people see clearly.
Cameras, telescopes, and microscopes: Use lenses to capture or magnify images.
Human eye: Has a convex lens that changes shape to focus on near (e.g., reading) or far objects.
Key Points to Remember
Concave Mirror Images: A concave mirror forms images that can be bigger (enlarged), smaller (diminished), or the same size as the object. The image can be upright (erect) or upside down (inverted), all depending on how far the object is from the mirror.
Convex Mirror Images: A convex mirror always forms images that are upright (erect) and smaller (diminished) than the object, no matter the distance.
Laws of Reflection: These are two key rules for how light bounces off mirrors: (1) The angle where the light hits (angle of incidence) is equal to the angle where it bounces back (angle of reflection). (2) The incoming light ray, the normal line (straight up from the mirror at the hit point), and the bouncing ray all stay in the same flat surface (plane).
Validity of Reflection Laws: The laws of reflection work the same way for every type of mirror, whether it’s flat (plane), curved inward (concave), or curved outward (convex).
Behavior of Mirrors with Light: A concave mirror brings light rays together (converges them), like focusing sunlight to make heat. A convex mirror spreads light rays apart (diverges them), giving a wider view.
Convex Lens Images: A convex lens forms images that can be bigger (enlarged), smaller (diminished), or the same size as the object. The image can be upright (erect) or upside down (inverted), depending on how far the object is from the lens.
Concave Lens Images: A concave lens always forms images that are upright (erect) and smaller (diminished) than the object, no matter the distance.
Behavior of Lenses with Light: A convex lens brings light rays together (converges them), like in a magnifying glass. A concave lens spreads light rays apart (diverges them), like in some eyeglasses.
Q1. What havoc has the super cyclone wreaked in the life of the people of Orissa? Ans. The super cyclone was a highly destructive storm that washed away houses and destroyed hundreds of villages. It killed thousands of people and uprooted a large number of trees. There were dead bodies everywhere, leaving many people homeless; several children lost their parents and became orphans. This devastation continued for the next thirty-six hours, and the overall condition of the affected areas was extremely grim.
Weathering Storm in Ersama
Q2. How has Prashant, a teenager, been able to help the people of his village? Ans. Prashant stepped forward as a young leader and organised both youths and elders to work together for relief. He and the group pressed the merchants to supply rice to people living in the shelters. He formed a team of youth volunteers who cleaned the shelters and tended to the wounds and fractures of those injured. Prashant then brought the orphaned children together, provided a polythene shelter for them and organised sports and activities to keep the children occupied and hopeful. He also helped mobilise women to look after the orphans while men secured food and other essentials for the shelter.
Q3. How have the people of the community helped one another? What role do the women of Kalikuda play during these days? Ans.The people of the community joined hands and began relief work under Prashant’s initiative; they helped one another by sharing tasks and supporting those most in need. The women of Kalikuda took active part in the relief efforts: they worked in the food-for-work programme and looked after the orphaned children, providing care and comfort during a very difficult time.
Q4. Why do Prashant and other volunteers resist the plan to set up institutions for orphans and widows? What alternatives do they consider? Ans. Prashant and the volunteers opposed the idea of setting up institutional homes because they believed such institutions would deprive children of family love and leave widows to suffer stigma and loneliness. Instead, they considered placing orphans within their own community or with foster families, such as childless widows or relatives, so that the children could grow up with family care and the widows would remain integrated in social life.
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Q5. Do you think Prashant is a good leader? Do you think young people can get together to help people during natural calamities? Ans. Yes, Prashant shows the qualities of a good leader. He takes initiative, has a clear vision, faces adverse circumstances courageously and motivates others by his energy and example. Young people possess great strength and enthusiasm, and when they come together they can provide effective help during natural calamities. Given responsibility, youth are capable of performing their duties with full spirit and making a real difference.
Q1. Give three examples of data which you can get from your day-to-day life. Sol: Here are the three examples which are related to our day-to-day life:
The number of boys in a sports team.
Electricity bills for last one year.
The number of students appearing for board exams at your school.
Q2: The number of family members in 10 flats of society are 2, 4, 3, 3, 1,0,2,4,1,5. Find the mean number of family members per flat. Sol: Number of family members in 10 flats -2, 4, 3, 3, 1, 0, 2, 4, 1, 5. So, we get, Mean = sum of observation/ total no of observations Mean = (2 + 4+ 3 + 3 + 1 + 0 + 2 + 4 + 1 + 5) / 10 Mean = 25/10 = 2.5
Q3: The daily minimum questions solved by a student during a week were as under:
Find the mean. Sol: Number of questions solved in a week: 35, 30, 27, 32, 23, 28. So, we get, Mean = sum of observation/ total no of observations = (35+30+27+32+23+28) / 6 = 175/6 = 29.167
Q4: The mean weight of a class of 34 students is 46.5 kg. If the weight of the new boy is included, the mean is rises by 500 g. Find the weight of the new boy. Sol: The mean weight of 34 students = 46.5 Sum of the weight of 34 students = (46.5 * 34) = 1581 Change or increase in the mean weight when the weight of a new boy is added = 0.5 So, the new mean = (46.5 +0.5) = 47 So, let the weight of the new boy be y. So, (sum of weight of 34 students + weight of new boy) / 35 = 47 (1581+ y)/ 35 = 47 1581 + y = 1645 y = 1645 – 1581 = 64
Q5: The height of 20 students of class V are noted as follows
Make a frequency distribution table for the above data.
Which is the most common height and which is the rarest height among these students?
Sol:
1. The required frequency distribution table is:
2. The most common heights are 4 and 5.5. The rarest heights are 4.6 and 4.7.
Q6: The following is the list of number of coupons issued in a school canteen during a week: 105, 216, 322, 167, 273, 405 and 346. Find the average no. of coupons issued per day. Sol: Number of coupons issued in a week: 105, 216, 322, 167, 273, 405 and 346. So, we get, Mean = sum of observation/ total no of observations Mean = (106+ 215+ 323+166+273+405+346)/ 7 = 1834/7 Mean = 262
Q7: If the mean of six observations y, y + 1, y + 4, y + 6, y + 8, y + 5 is 13, find the value of y. Sol: Mean = sum of observation/ total no of observations 13 = (y + y + 1+ y + 4+ y + 6+ y + 8+ y + 5) / 6 13 = (6y + 24)/6 (13 * 6) = 6y +24 (13 * 6) – 24 = 6y (13 * 6) – 6 * 4 = 6y 6(13 – 4) = 6y y = 9
Q8: The Number of books issued to 13 students in a class are: 25, 19, 24, 23, 29, 31, 19, 20, 22, 26, 17, 35, 21. Find the median no. of books for the above data. Sol: Let’s arrange the data given in ascending order – 17, 19, 19, 20, 21, 22, 23, 24, 25, 26, 29,31,35. n= 13, so it’s an odd number Median = (n+1) / 2 observations = (13+1)/ 2 = (14/2)th observation = 7th observation = 23
Q9: The weight (in kg) of 7 students of a class are 44, 52, 55, 60, 50, 49, 45. Find the median weight. Sol: Let’s arrange the data given in ascending order – 44, 45, 49, 50, 52, 55, 60. n= 7, so it’s an odd number Median = (n+1) / 2 observations = (7+1)/ 2 = (8/2)th observation = 4th observation = 50 kg
Q1: In a hot water heating system, there is a cylindrical pipe of length 28 m and diameter 5 cm. Find the total radiating surface in the system. Sol: Given, Length of the cylindrical pipe = h = 28 m Diameter of the pipe = 5 cm Now, the radius of piper (r) = 5/ 2 cm = 2.5 cm = 0.025 m Total radiating surface in the system = Total surface area of the cylinder = 2πr(h + r) = 2 × (22/7) × 0.025 (28 + 0.025) m2 = (44 x 0.025 x 28.025)/7 m2 = 4.4 m2 (approx)
Q2: Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m. Sol: Given, Diameter of the cone = 24 m Radius of the cone (r) = 24/2 = 12 m Slant height of the cone (l) = 21 m Total surface area of a cone = πr(l + r) = (22/7) × 12 × (21 + 12) = (22/7) × 12 × 33 = 1244.57 m2
Q3: The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding. Sol: Given, Diameter of the sphere = 7 m Radius (r) = 7/2 = 3.5 m Now, the riding space available for the motorcyclist = Surface area of the sphere = 4πr2 = 4 × (22/7) × 3.5 × 3.5 = 154 m2
Q4: Hameed has built a cubical water tank with a lid for his house, with each outer edge 1.5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm (see in the figure below). Find how much he would spend on the tiles if the cost of the tiles is Rs.360 per dozen.
Sol: Given, Edge of the cubical tank (a) = 1.5 m = 150 cm So, surface area of the tank = 5 × 150 × 150 cm2 The measure of side of a square tile = 25 cm Area of each square tile = side × side = 25 × 25 cm2 Required number of tiles = (Surface area of the tank)/(area of each tile) = (5 × 150 × 150)/(25 × 25) = 180 Also, given that the cost of the tiles is Rs. 360 per dozen. Thus, the cost of each tile = Rs. 360/12 = Rs. 30 Hence, the total cost of 180 tiles = 180 × Rs. 30 = Rs. 5400
Q5: The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of whitewashing the walls of the room and the ceiling at the rate of Rs.7.50 per sq.m. Sol: Given, Length of the room (l) = 5 m Breadth of the room (b) = 4 m Height of the room (h) = 3 m Area of walls of the room = Lateral surface area of cuboid = 2h(l + b) = 2 × 3(5 + 4) = 6 × 9 = 54 sq.m Area of ceiling = Area of base of the cuboid = lb = 5 × 4 = 20 sq.m Area to be white washed = (54 + 20) sq.m = 74 sq.m Given that, the cost of white washing 1 sq.m = Rs. 7.50 Therefore, the total cost of white washing the walls and ceiling of the room = 74 × Rs. 7.50 = Rs. 555
Q6: Curved surface area of a right circular cylinder is 4.4 sq.m. If the radius of the base of the cylinder is 0.7 m, find its height. Sol: Let h be the height of the cylinder. Given, Radius of the base of the cylinder (r) = 0.7 m Curved surface area of cylinder = 4.4 m2 Thus, 2πrh = 4.4 2 × 3.14 × 0.7 × h = 4.4 4.4 × h = 4.4 h = 4.4/4.4 h = 1 Therefore, the height of the cylinder is 1 m.
Q7: The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone. (Take π = 3.14) Sol: Given Height of a cone (h) = 16 cm Radius of the base (r) = 12 cm Now, Slant height of cone (l) = √(r2 + h2) = √(256 + 144) = √400 = 20 cm Curved surface area of cone = πrl = 3.14 × 12 × 20 cm2 = 753.6 cm2 Total surface area = πrl + πr2 = (753.6 + 3.14 × 12 × 12) cm2 = (753.6 + 452.16) cm2 = 1205.76 cm2
Q8: The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs.210 per 100 sq.m. Sol: Given, Slant height of a cone (l) = 25 m Diameter of the base of cone = 2r = 14 m ∴ Radius = r = 7 m Curved Surface Area = πrl = (22/7) x 7 x 25 = 22 × 25 = 550 sq.m Also, given that the cost of white-washing 100 sq.m = Rs. 210 Hence, the total cost of white-washing for 550 sq.m = (Rs. 210 × 550)/100 = Rs. 1155
Q9: The paint in a certain container is sufficient to paint an area equal to 9.375 sq.m. How many bricks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container? Sol: Given, Dimensions of the brick = 22.5 cm × 10 cm × 7.5 cm Here, l = 22.5 cm, b = 10 cm, h = 7.5 cm Surface area of 1 brick = 2(lb + bh + hl) = 2(22.5 × 10 + 10 × 7.5 + 7.5 × 22.5) cm2 = 2(225 + 75 + 168.75) cm2 = 2 x 468.75 cm2 = 937.5 cm2 Area that can be painted by the container = 9.375 m2 (given) = 9.375 × 10000 cm2 = 93750 cm2 Thus, the required number of bricks = (Area that can be painted by the container)/(Surface area of 1 brick) = 93750/937.5 = 937500/9375 = 100
Q10: The curved surface area of a right circular cylinder of height 14 cm is 88 sq.cm. Find the diameter of the base of the cylinder. Sol: Let d be the diameter and r be the radius of a right circular cylinder. Given, Height of cylinder (h) = 14 cm Curved surface area of right circular cylinder = 88 cm2 ⇒ 2πrh = 88 cm2 ⇒ πdh = 88 cm2 (since d = 2r) ⇒ 22/7 x d x 14 cm = 88 cm2 ⇒ d = 2 cm Hence, the diameter of the base of the cylinder is 2 cm.
Q1: Find the area of a triangle whose two sides are 18 cm and 10 cm and the perimeter is 42cm. Sol: Assume that the third side of the triangle to be “x”. Now, the three sides of the triangle are 18 cm, 10 cm, and “x” cm It is given that the perimeter of the triangle = 42cm So, x = 42 – (18 + 10) cm = 14 cm ∴ The semi perimeter of triangle = 42/2 = 21 cm Using Heron’s formula,
Q2: A field is in the shape of a trapezium whose parallel sides are 25 m and 10 m. The non-parallel sides are 14 m and 13 m. Find the area of the field. Sol: First, draw a line segment BE parallel to the line AD. Then, from B, draw a perpendicular on the line segment CD. Now, it can be seen that the quadrilateral ABED is a parallelogram. So,AB = ED = 10 m AD = BE = 13 m EC = 25 – ED = 25 – 10 = 15 m Now, consider the triangle BEC, Its semi perimeter (s) = (13+ 14 + 15)/2 = 21 m By using Heron’s formula, Area of ΔBEC = = √[s(s − a)(s − b)(s − c)] = √(21 × (21 − 13) × (21 − 14) × (21 − 15)) m2 = √(21 × 8 × 7 × 6) m2 = 84 m2 We also know that the area of ΔBEC = (½) × CE × BF 84 cm2 = (½) × 15 × BF ⇒ BF = (168/15) cm = 11.2 cm So, the total area of ABED will be BF × DE, i.e. 11.2 × 10 = 112 m2 ∴ Area of the field = 84 + 112 = 196 m2
Q3: Find the cost of laying grass in a triangular field of sides 50 m, 65 m and 65 m at the rate of Rs 7 per m2. Sol: According to the question, Sides of the triangular field are 50 m, 65 m and 65 m. Cost of laying grass in a triangular field = Rs 7 per m2 Let a = 50, b = 65, c = 65 s = (a + b + c)/2 ⇒ s = (50 + 65 + 65)/2 = 180/2 = 90. Area of triangle = √(s(s-a)(s-b)(s-c)) = √(90(90 – 50)(90 – 65)(90 – 65)) = √(90 × 40 × 25 × 25) = 1500m2 Cost of laying grass = Area of triangle × Cost per m2 = 1500 × 7 = Rs.10500
Q4: How much paper of each shade is needed to make a kite given in the figure, in which ABCD is a square with diagonal 44 cm.
Sol: According to the figure, AC = BD = 44cm AO = 44/2 = 22cm BO = 44/2 = 22cm From ΔAOB, AB2 = AO2 + BO2 ⇒ AB2 = 222 + 222 ⇒ AB2 = 2 × 222 ⇒ AB = 22√2 cm Area of square = (Side)2 = (22√2)2 = 968 cm2 Area of each triangle (I, II, III, IV) = Area of square /4 = 968 /4 = 242 cm2 To find area of lower triangle, Let a = 20, b = 20, c = 14 s = (a + b + c)/2 ⇒ s = (20 + 20 + 14)/2 = 54/2 = 27. Area of the triangle = √[s(s-a)(s-b)(s-c)] = √[27(27 – 20)(27 – 20)(27 – 14)] = √[27 × 7 × 7 × 13] = 131.14 cm2 Therefore, We get, Area of Red = Area of IV = 242 cm2 Area of Yellow = Area of I + Area of II = 242 + 242 = 484 cm2 Area of Green = Area of III + Area of the lower triangle = 242 + 131.14 = 373.14 cm2
Q5: The sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540cm. Find its area. Sol: The ratio of the sides of the triangle is given as 12: 17: 25 Now, let the common ratio between the sides of the triangle be “x” ∴ The sides are 12x, 17x and 25x It is also given that the perimeter of the triangle = 540 cm 12x + 17x + 25x = 540 cm ⇒ 54x = 540cm So, x = 10 Now, the sides of the triangle are 120 cm, 170 cm, 250 cm. So, the semi perimeter of the triangle (s) = 540/2 = 270 cm Using Heron’s formula, Area of the triangle = √[s(s − a)(s − b)(s − c)] = √(270(270 − 120)(270 − 170)(270 − 250)) cm2 = √(270 × 150 × 100 × 20) cm2 = 9000 cm2
Q6: A rhombus-shaped field has green grass for 18 cows to graze. If each side of the rhombus is 30 m and its longer diagonal is 48 m, how much area of grass field will each cow be getting? Sol: Draw a rhombus-shaped field first with the vertices as ABCD. The diagonal AC divides the rhombus into two congruent triangles which are having equal areas. The diagram is as follows.
Consider the triangle BCD, Its semi-perimeter = (48 + 30 + 30)/2 m = 54 m Using Heron’s formula, Area of the ΔBCD = = √[s(s − a)(s − b)(s − c)] = √(54(54 − 48)(54 − 30)(54 − 30)) m2 = √(54 × 6 × 24 × 24) m2 = 432 m2 ∴ Area of field = 2 × area of the ΔBCD = (2 × 432) m2 = 864 m2 Thus, the area of the grass field that each cow will be getting = (864/18) m2 = 48 m2
Q7: The perimeter of an isosceles triangle is 32 cm. The ratio of the equal side to its base is 3: 2. Find the area of the triangle. Sol: According to the question, The perimeter of the isosceles triangle = 32 cm It is also given that, Ratio of equal side to base = 3 : 2 Let the equal side = 3x So, base = 2x Perimeter of the triangle = 32 ⇒ 3x + 3x + 2x = 32 ⇒ 8x = 32 ⇒ x = 4. Equal side = 3x = 3×4 = 12 Base = 2x = 2×4 = 8 The sides of the triangle = 12cm, 12cm and 8cm. Let a = 12, b = 12, c = 8 s = (a + b + c)/2 ⇒ s = (12 + 12 + 8)/2 = 32/2 = 16. Area of the triangle = √(s(s – a)(s – b)(s – c)) = √(16(16 – 12)(16 – 12)(16 – 8)) = √(16 × 4 × 4 × 8) = 32√2 cm2
Q8: A rectangular plot is given for constructing a house, having a measurement of 40 m long and 15 m in the front. According to the laws, a minimum of 3 m, wide space should be left in the front and back each and 2 m wide space on each of other sides. Find the largest area where a house can be constructed. Sol:
Let the given rectangle be rectangle PQRS, According to the question, PQ = 40m and QR = 15m As 3m is left in both front and back, AB = PQ -3 -3 ⇒ AB = 40 -6 ⇒ AB = 34m Also, Given that 2m has to be left at both the sides, BC = QR -2 – 2 ⇒ BC = 15 -4 ⇒ BC = 11m Now, Area left for house construction is the area of ABCD. Hence, Area(ABCD) = AB × CD = 34 × 11 = 374 m2
Q9: Find the area of a triangle whose sides are respectively 150 cm, 120 cm, and 200 cm?
Sol: The triangle whose sides are: a = 150 cm, b = 120 cm, c = 200 cm The area of a triangle = √[s(s − a)(s − b)(s − c)] Here, s = semi-perimeter of the triangle 2s = a + b + c s = a + b + c2 = 150 + 200 + 1202 = 235 cm Area of triangle = √[s(s − a)(s − b)(s − c)] = √(235(235 − 150)(235 − 200)(235 − 120)) cm2 = √(235 × 85 × 35 × 115) cm2 = 8966.56 cm2
Q10: A triangular park has a perimeter of 300 m, and all its sides are equal in length. Find the area of the park using Heron’s formula. Sol: Let each side of the equilateral triangle be a. The semi-perimeter of the triangle,
s = a + a + a2 = 3a2 Using Heron’s formula: Area = √[s(s − a)(s − b)(s − c)] = √[s(s − a)3] Area = √[ 3a2 × 3a2 − a ]3 Area = √34 a2 Now, the perimeter = 300 m a + a + a = 300 ⇒ 3a = 300 ⇒ a = 100 m Thus, area of park = √34 (100)2 = 2500√3 m2
Q1: Bisectors of angles A, B and C of a triangle ABC intersect its circumcircle at D, E and F respectively. Prove that the angles of the triangle DEF are 90° – (½)A, 90° – (½)B and 90° – (½)C. Sol: Consider the following diagram: Here, ABC is inscribed in a circle with center O and the bisectors of ∠A, ∠B and ∠C intersect the circumcircle at D, E and F respectively. Now, join DE, EF and FD As angles in the same segment are equal, so, ∠FDA = ∠FCA ————-(i) ∠FDA = ∠EBA ————-(i) Adding equations (i) and (ii) we have, ∠FDA + ∠EDA = ∠FCA + ∠EBA Or, ∠FDE = ∠FCA + ∠EBA = (½)∠C + (½)∠B We know, ∠A + ∠B + ∠C = 180° So, ∠FDE = (½)[∠C + ∠B] = (½)[180° – ∠A] ⇒ ∠FDE = [90 – (∠A/2)] In a similar way, ∠FED = [90 – (∠B/2)] And, ∠EFD = [90 – (∠C/2)]
Q2: Prove that the circle drawn with any side of a rhombus as diameter passes through the point of intersection of its diagonals. Sol: To prove: A circle drawn with Q as centre, will pass through A, B and O (i.e. QA = QB = QO) Since all sides of a rhombus are equal, AB = DC Now, multiply (½) on both sides (½)AB = (½)DC So, AQ = DP ⇒ BQ = DP Since Q is the midpoint of AB, AQ= BQ Similarly, RA = SB Again, as PQ is drawn parallel to AD, RA = QO Now, as AQ = BQ and RA = QO we have, QA = QB = QO (hence proved).
Q3: If circles are drawn taking two sides of a triangle as diameters, prove that the point of intersection of these circles lies on the third side. Sol: First, draw a triangle ABC and then two circles having a diameter as AB and AC respectively. We will have to now prove that D lies on BC and BDC is a straight line. Proof: As we know, angle in the semi-circle are equal So, ∠ADB = ∠ADC = 90° Hence, ∠ADB + ∠ADC = 180° ∴ ∠BDC is a straight line. So, it can be said that D lies on the line BC.
Q4: ABCD is a cyclic quadrilateral whose diagonals intersect at a point E. If ∠DBC = 70°, ∠BAC is 30°, find ∠BCD. Further, if AB = BC, find ∠ECD. Sol: Consider the following diagram. Consider the chord CD, As we know, angles in the same segment are equal. So, ∠CBD = ∠CAD ∴ ∠CAD = 70° Now, ∠BAD will be equal to the sum of angles BAC and CAD. So, ∠BAD = ∠BAC + ∠CAD = 30° + 70° ∴ ∠BAD = 100° As we know, the opposite angles of a cyclic quadrilateral sum up to 180 degrees. So, ∠BCD + ∠BAD = 180° Since, ∠BAD = 100° So, ∠BCD = 80° Now consider the ΔABC. Here, it is given that AB = BC Also, ∠BCA = ∠CAB (Angles opposite to equal sides of a triangle) ∠BCA = 30° also, ∠BCD = 80° ∠BCA + ∠ACD = 80° So, ∠ACD = 50° and, ∠ECD = 50°
Q5: In Figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR. Sol: Since angle which is subtended by an arc at the centre of the circle is double the angle subtended by that arc at any point on the remaining part of the circle. So, the reflex ∠POR = 2 × ∠PQR We know the values of angle PQR as 100° So, ∠POR = 2 × 100° = 200° ∴ ∠POR = 360° – 200° = 160° Now, in ΔOPR, OP and OR are the radii of the circle So, OP = OR Also, ∠OPR = ∠ORP Now, we know sum of the angles in a triangle is equal to 180 degrees So, ∠POR + ∠OPR + ∠ORP = 180° ⇒ ∠OPR + ∠OPR = 180° – 160° As ∠OPR = ∠ORP ⇒ 2∠OPR = 20° Thus, ∠OPR = 10°
Q6: In any triangle ABC, if the angle bisector of ∠A and perpendicular bisector of BC intersect, prove that they intersect on the circumcircle of the triangle ABC. Sol: Consider this diagram: Given: In ∆ABC, AD is the angle bisector of ∠A and OD is the perpendicular bisector of BC, intersecting each other at point D. To Prove: D lies on the circle Construction: Join OB and OC Proof: BC is a chord of the circle. The perpendicular bisector will pass through centre O of the circumcircle. ∴ OE ⊥ BC & E is the midpoint of BC Chord BC subtends twice the angle at the centre, as compared to any other point. BC subtends ∠BAC on the circle & BC subtends ∠BOC on the centre ∴ ∠BAC = 1/2 ∠ BOC In ∆ BOE and ∆COE, BE = CE (OD bisects BC) ∠BEO = ∠CEO (Both 90°, as OD ⊥ BC) OE = OE (Common) ∴ ∆BOE ≅ ∆COE (SAS Congruence rule) ∴ ∠BOE = ∠COE (CPCT) Now, ∠BOC = ∠BOE + ∠COE ∠BOC = ∠BOE + ∠BOE ∠BOC = 2 ∠BOE …(2) AD is angle bisector of ∠A ∴ ∠BAC = 2∠BAD From (1) ∠BAC = 1/2 ∠BOC 2 ∠BAD = 1/2 (2∠BOE) 2 ∠BAD = ∠BOE ∠BAD = 1/2 ∠BOE BD subtends ∠BOE at centre and half of its angle at Point A. Hence, BD must be a chord. ∴ D lies on the circle.
Q7: Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6, find the radius of the circle. Sol: Here, OM ⊥ AB and ON ⊥ CD. is drawn and OB and OD are joined. As we know, AB bisects BM as the perpendicular from the centre bisects the chord. Since AB = 5 so, BM = AB/2 Similarly, ND = CD/2 = 11/2 Now, let ON be x. So, OM = 6− x. Consider ΔMOB, OB2 = OM2 + MB2 Or, OB2 = 36 + x2 – 12x + 25/4 ……(1) Consider ΔNOD, OD2 = ON2 + ND2 Or, OD2 = x2 + 121/4 ……….(2) We know, OB = OD (radii) From eq. (1) and eq. (2) we have; 36 + x2 -12x + 25/4 = x2 + 121/4 12x = 36 + 25/4 – 121/4 12x = (144 + 25 -121)/4 12x = 48/4 = 12 x = 1 Now, from eq. (2) we have, OD2 = 11 + (121/4) Or OD = (5/2) × √5
Q8: If the non-parallel sides of a trapezium are equal, prove that it is cyclic. Sol: Construction-Consider a trapezium ABCD with AB||CD and BC = AD. Draw AM ⊥CD and BN ⊥ CD In ∆AMD and ∆BNC; AD = BC (Given) ∠AMD = ∠BNC (90°) AM =BN (perpendiculars between parallel lines) ∆AMD = ∆BNC (By RHS congruency) ∆ADC = ∆BCD (By CPCT rule) …….(i) ∠BAD and ∠ADC are on the same side of transversal AD. ∠BAD + ∠ADC = 180° ……(ii) ∠BAD + ∠BCD = 180° (by equation (i)) Since, the opposite angles are supplementary, therefore, ABCD is a cyclic quadrilateral.
As we know, angles in the segment of the circle are equal so, ∠BAC = ∠BDC Now in the In ΔABC, sum of all the interior angles will be 180° So, ∠ABC + ∠BAC + ∠ACB = 180° Now, by putting the values, ∠BAC = 180° – 69° – 31° So, ∠BAC = 80°
Q10: A circular park of radius 20m is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in his hands to talk each other. Find the length of the string of each phone. Sol: First, draw a diagram according to the given statements. The diagram will look as follows. Here the positions of Ankur, Syed and David are represented as A, B and C respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle. AD ⊥ BC is drawn. Now, AD is median of ΔABC and it passes through the centre O. Also, O is the centroid of the ΔABC. OA is the radius of the triangle. OA = 2/3 AD Let the side of a triangle a metres then BD = a/2 m. Applying Pythagoras theorem in ΔABD, AB2 = BD2 + AD2 ⇒ AD2 = AB2 – BD2 ⇒ AD2 = a2 – (a/2)2 ⇒ AD2 = 3a2/4 ⇒ AD = √3a/2 OA = 2/3 AD ⇒ 20 m = 2/3 × √3a/2 ⇒ a = 20√3 m So, the length of the string of the toy is 20√3 m.