The Triangle and Its Properties | Grade 7th - Mathematics Chapter Summary & NCERT CBSE Study Resources

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7th

7th - Mathematics

The Triangle and Its Properties

Overview of the Chapter: The Triangle and Its Properties

This chapter introduces students to the fundamental concepts related to triangles, their types, properties, and important theorems. It covers the basics of triangles, including their sides, angles, and the relationship between them, as per the CBSE Grade 7 Mathematics curriculum.

1. Introduction to Triangles

A triangle is a simple closed curve made of three line segments. It has three vertices, three sides, and three angles.

A triangle is a polygon with three edges and three vertices.

2. Types of Triangles

Triangles can be classified based on their sides and angles:

Based on sides: Equilateral, Isosceles, Scalene
Based on angles: Acute-angled, Right-angled, Obtuse-angled

3. Properties of Triangles

Key properties include:

The sum of the interior angles of a triangle is always 180°.
The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

The angle sum property states that the sum of the three interior angles of a triangle is 180°.

4. Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

For a right-angled triangle with sides a, b, and hypotenuse c: a² + b² = c².

5. Medians and Altitudes of a Triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. An altitude is a perpendicular line segment from a vertex to the line containing the opposite side.

6. Equilateral and Isosceles Triangles

An equilateral triangle has all sides equal and all angles equal to 60°. An isosceles triangle has two sides equal and the angles opposite the equal sides are also equal.

7. Sum of the Lengths of Two Sides of a Triangle

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Triangle inequality theorem: For any triangle, the sum of any two sides must be greater than the third side.

8. Right-Angled Triangles and Pythagoras Property

Right-angled triangles follow the Pythagoras property, which is useful in determining the length of one side if the other two are known.

All Question Types with Solutions – CBSE Exam Pattern

Explore a complete set of CBSE-style questions with detailed solutions, categorized by marks and question types. Ideal for exam preparation, revision and practice.

Very Short Answer (1 Mark) – with Solutions (CBSE Pattern)

These are 1-mark questions requiring direct, concise answers. Ideal for quick recall and concept clarity.

Question 1:

What is the sum of the angles in a triangle?

Answer:

180°

Question 2:

Name the triangle with all sides equal.

Answer:

Equilateral triangle

Question 3:

If two angles of a triangle are 45° and 45°, what is the third angle?

Answer:

90°

Question 4:

What is the Pythagorean theorem for a right-angled triangle?

Answer:

Hypotenuse² = Base² + Height²

Question 5:

A triangle has sides 3 cm, 4 cm, and 5 cm. What type of triangle is it?

Answer:

Right-angled triangle

Question 6:

What is the exterior angle property of a triangle?

Answer:

Exterior angle = Sum of two opposite interior angles

Question 7:

If one angle of a triangle is 90°, what is it called?

Answer:

Right-angled triangle

Question 8:

Can a triangle have two right angles?

Answer:

Question 9:

What is the median of a triangle?

Answer:

Line joining vertex to midpoint of opposite side

Question 10:

In an isosceles triangle, how many sides are equal?

Answer:

Two

Question 11:

If a triangle has angles 60°, 60°, and 60°, what type is it?

Answer:

Equilateral triangle

Question 12:

What is the shortest side opposite to in a triangle?

Answer:

The smallest angle

Question 13:

Define a triangle.

Answer:

A triangle is a three-sided polygon formed by three line segments joining three non-collinear points. It has three vertices, three sides, and three angles.

Question 14:

What is the sum of the interior angles of a triangle?

Answer:

The sum of the interior angles of a triangle is always 180 degrees. This is a fundamental property known as the Angle Sum Property.

Question 15:

Name the type of triangle where all sides are equal.

Answer:

A triangle with all sides equal is called an equilateral triangle. In such a triangle, all three angles are also equal and measure 60 degrees each.

Question 16:

What is the Pythagorean theorem?

Answer:

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Formula: Hypotenuse² = Base² + Height².

Question 17:

Identify the triangle with two sides equal.

Answer:

A triangle with two sides equal is called an isosceles triangle. The angles opposite the equal sides are also equal.

Question 18:

Calculate the third angle of a triangle if two angles are 45° and 45°.

Answer:

Using the Angle Sum Property:
Sum of angles = 180°
Given angles = 45° + 45° = 90°
Third angle = 180° - 90° = 90°.

Question 19:

What is a scalene triangle?

Answer:

A scalene triangle is a triangle where all three sides have different lengths, and all three angles are of different measures.

Question 20:

State the Triangle Inequality Property.

Answer:

The Triangle Inequality Property states that the sum of any two sides of a triangle must be greater than the third side.
For sides a, b, c: a + b > c, b + c > a, and a + c > b.

Question 21:

If one angle of a triangle is 90°, what type of triangle is it?

Answer:

A triangle with one angle measuring 90° is called a right-angled triangle. The side opposite the right angle is the hypotenuse, which is the longest side.

Very Short Answer (2 Marks) – with Solutions (CBSE Pattern)

These 2-mark questions test key concepts in a brief format. Answers are expected to be accurate and slightly descriptive.

Question 1:

Define a triangle and list its three basic elements.

Answer:

A triangle is a three-sided polygon formed by connecting three non-collinear points. Its three basic elements are:

Sides (the three line segments)
Vertices (the three corner points)
Angles (the three angles formed at the vertices)

Question 2:

What is the angle sum property of a triangle?

Answer:

The angle sum property states that the sum of the three interior angles of any triangle is always 180 degrees.

For example, if angles are 60°, 70°, and 50°:
60° + 70° + 50° = 180°.

Question 3:

Identify the type of triangle where all sides are equal.

Answer:

A triangle with all sides equal is called an equilateral triangle.

In such triangles:
All three angles are also equal (each 60°).
It has three lines of symmetry.

Question 4:

State the Pythagorean theorem for a right-angled triangle.

Answer:

The Pythagorean theorem states that in a right-angled triangle:
Hypotenuse² = Base² + Perpendicular².

For example, if base = 3 cm, perpendicular = 4 cm:
Hypotenuse² = 3² + 4² = 9 + 16 = 25
Hypotenuse = √25 = 5 cm.

Question 5:

What is an exterior angle of a triangle and how is it related to its interior opposite angles?

Answer:

An exterior angle is formed by extending one side of the triangle.

It is equal to the sum of the two interior opposite angles.
For example, if interior opposite angles are 50° and 60°:
Exterior angle = 50° + 60° = 110°.

Question 6:

Differentiate between a scalene and an isosceles triangle.

Answer:

Scalene triangle:
All sides are of different lengths.
All angles are of different measures.

Isosceles triangle:
Two sides are equal in length.
Angles opposite to equal sides are equal.

Question 7:

Explain why a triangle cannot have two right angles.

Answer:

If a triangle had two right angles (90° each), their sum would be 180°.
This leaves no room for a third angle (since total must be 180°), which violates the definition of a triangle.

Question 8:

Find the perimeter of an equilateral triangle with each side measuring 5 cm.

Answer:

Perimeter = Sum of all sides
For equilateral triangle:
Perimeter = 5 cm + 5 cm + 5 cm = 15 cm.

Question 9:

In a right-angled triangle, if one acute angle is 30°, what is the measure of the other acute angle?

Answer:

Sum of angles in a triangle = 180°
Right angle = 90°
Given acute angle = 30°
Other acute angle = 180° - (90° + 30°) = 60°.

Short Answer (3 Marks) – with Solutions (CBSE Pattern)

These 3-mark questions require brief explanations and help assess understanding and application of concepts.

Question 1:

Define a scalene triangle and list its properties.

Answer:

A scalene triangle is a type of triangle where all three sides have different lengths, and all three angles are of different measures.

All sides are unequal.
All angles are unequal.
No line of symmetry exists.

Example: A triangle with sides 5 cm, 6 cm, and 7 cm is scalene.

Question 2:

Explain the Pythagorean theorem with an example.

Answer:

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Formula: Hypotenuse² = Base² + Perpendicular²

Example: For a right-angled triangle with sides 3 cm and 4 cm,
Hypotenuse² = 3² + 4² = 9 + 16 = 25
Hypotenuse = √25 = 5 cm.

Question 3:

What is the sum of interior angles of a triangle? Prove it using an activity.

Answer:

The sum of interior angles of a triangle is always 180°.

Activity to prove:
1. Draw any triangle (e.g., ΔABC).
2. Cut out the triangle and tear off its three angles.
3. Arrange the three angles side by side.
4. You will observe they form a straight line (180°).

This proves that the sum of the interior angles of a triangle is 180°.

Question 4:

Differentiate between an acute-angled triangle and an obtuse-angled triangle.

Answer:

Acute-angled triangle:

All three angles are less than 90°.
Example: A triangle with angles 60°, 70°, and 50°.

Obtuse-angled triangle:

One angle is greater than 90°.
The other two angles are acute (less than 90°).
Example: A triangle with angles 100°, 40°, and 40°.

Question 5:

Calculate the third angle of a triangle if two angles are 45° and 55°.

Answer:

Given two angles: 45° and 55°.
Let the third angle be x.

We know the sum of angles in a triangle is 180°.
So, 45° + 55° + x = 180°
100° + x = 180°
x = 180° - 100°
x = 80°

Thus, the third angle is 80°.

Question 6:

Differentiate between an acute-angled and an obtuse-angled triangle.

Answer:

Acute-angled triangle: All three angles are less than 90°.
Obtuse-angled triangle: One angle is greater than 90°.

Key difference:
In an acute-angled triangle, no angle reaches 90°, while an obtuse-angled triangle has one angle exceeding 90°.

Question 7:

Calculate the third angle of a triangle if two angles are 45° and 65°.

Answer:

Given two angles: 45° and 65°.

Step 1: Sum of given angles = 45° + 65° = 110°.
Step 2: Sum of all angles in a triangle = 180°.
Step 3: Third angle = 180° - 110° = 70°.

The third angle is 70°.

Long Answer (5 Marks) – with Solutions (CBSE Pattern)

These 5-mark questions are descriptive and require detailed, structured answers with proper explanation and examples.

Question 1:

Explain the Pythagorean theorem with an example from our textbook and a real-life application.

Answer:

Introduction

We studied the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

Argument 1

Our textbook shows an example: A triangle with sides 3 cm, 4 cm, and 5 cm. Here, 5² = 3² + 4² (25 = 9 + 16), proving the theorem.

Argument 2

In real life, carpenters use this theorem to ensure corners are right-angled. For example, measuring 6 ft and 8 ft sides to check if the diagonal is 10 ft.

Conclusion

The theorem is useful in geometry and practical measurements.

Question 2:

Describe the properties of an equilateral triangle with a textbook example and a real-world use case.

Answer:

Introduction

An equilateral triangle has all sides equal and all angles measuring 60°.

Argument 1

Our textbook gives an example: A triangle with each side 5 cm and each angle 60°. This confirms the properties.

Argument 2

In real life, traffic signs like 'Yield' use equilateral triangles for uniformity and visibility.

Conclusion

Equilateral triangles are symmetrical and widely used in design and architecture.

Question 3:

Prove that the sum of angles in a triangle is 180° using a diagram and a textbook example.

Answer:

Introduction

We learned that the sum of angles in any triangle is always 180°.

Argument 1

[Diagram: A triangle ABC with angles A, B, and C labeled.] If we draw a line parallel to BC through A, angles B and C align with angles on the line, summing to 180°.

Argument 2

Our textbook shows a triangle with angles 60°, 70°, and 50°. Adding them (60° + 70° + 50° = 180°) verifies the property.

Conclusion

This property helps solve many geometry problems.

Question 4:

Describe the properties of an isosceles triangle with a diagram and an NCERT-based example.

Answer:

Introduction

An isosceles triangle has two equal sides and two equal angles opposite those sides.

Argument 1

[Diagram: Triangle ABC with AB = AC and ∠B = ∠C]. Our textbook gives an example: ΔABC with AB = AC = 5 cm and BC = 6 cm. Here, ∠B = ∠C = 70°.

Argument 2

We can verify this by drawing the altitude from A to BC, which divides it into two equal parts (3 cm each). This confirms symmetry.

Conclusion

Isosceles triangles are symmetrical and widely used in architecture, like roof designs.

Question 5:

Prove that the sum of angles in a triangle is 180° using an activity from NCERT and a step-wise derivation.

Answer:

Introduction

We learned that the sum of the three angles in any triangle is always 180°.

Argument 1

Our textbook shows an activity: Draw ΔABC, cut its angles, and arrange them on a straight line. They form 180°, proving the property.

Argument 2

Step-wise derivation:

Draw a line PQ parallel to BC through A.
∠PAB = ∠ABC (alternate angles).
∠QAC = ∠ACB (alternate angles).
∠PAB + ∠BAC + ∠QAC = 180° (straight line).

Conclusion

This property helps solve problems like finding missing angles in triangles.

Question 6:

Describe the properties of an isosceles triangle with a diagram and an NCERT example.

Answer:

Introduction

An isosceles triangle has two equal sides and two equal angles opposite those sides.

Argument 1

Our textbook gives an example: ΔABC with AB = AC = 5 cm and ∠B = ∠C = 70°.

Argument 2

[Diagram: Triangle with two equal sides and base angles marked.] The equal sides are called legs, and the third side is the base.

Conclusion

This property helps solve problems involving symmetry and congruence in triangles.

Question 7:

Prove that the sum of angles in a triangle is 180° using a practical activity and a textbook derivation.

Answer:

Introduction

We learned that the sum of the three angles in any triangle is always 180°.

Argument 1

Our textbook shows a proof by drawing a line parallel to one side, creating alternate angles that add up to 180°.

Argument 2

In a practical activity, we cut a paper triangle and arranged its angles to form a straight line, visually confirming the sum.

Conclusion

This property is foundational for solving geometric problems.

Question 8:

Explain the Pythagoras theorem with an example from our textbook and a real-life application.

Answer:

Introduction

We studied the Pythagoras theorem, which states that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

Argument 1

Our textbook shows an example: For a triangle with sides 3 cm, 4 cm, and hypotenuse 5 cm, 3² + 4² = 5² (9 + 16 = 25).

Argument 2

In real life, this theorem helps carpenters ensure corners are right-angled by measuring sides.

Conclusion

Thus, the theorem is useful in geometry and practical tasks.

Question 9:

Describe the properties of an equilateral triangle with a diagram and NCERT-based example.

Answer:

Introduction

An equilateral triangle has all sides equal and each angle measures 60°.

Argument 1

[Diagram: Triangle with sides AB = BC = CA and angles A = B = C = 60°]. Our textbook shows ∆ABC with AB = BC = CA = 5 cm.

Argument 2

It is used in real life, like traffic signs, due to its symmetry.

Conclusion

Thus, equilateral triangles are simple yet important in geometry.

Question 10:

Prove that the sum of angles in a triangle is 180° using an activity from NCERT.

Answer:

Introduction

We learned that the sum of angles in any triangle is always 180°.

Argument 1

Our textbook activity shows drawing a triangle, cutting its corners, and placing them together to form a straight line (180°).

Argument 2

This property helps architects design stable structures.

Conclusion

Hence, this rule is fundamental in geometry and construction.

Question 11:

In a triangle ABC, the measure of angle A is 50° and angle B is 70°. Find the measure of angle C. Also, identify the type of triangle based on its angles and justify your answer.

Answer:

To find the measure of angle C, we use the Angle Sum Property of a triangle, which states that the sum of all three interior angles of a triangle is 180°.

Given:
Angle A = 50°
Angle B = 70°

Step 1: Add the given angles.
Angle A + Angle B = 50° + 70° = 120°

Step 2: Subtract the sum from 180° to find Angle C.
Angle C = 180° - 120° = 60°

Now, to identify the type of triangle:

Since all three angles (50°, 70°, 60°) are less than 90°, the triangle is an acute-angled triangle.

Justification: An acute-angled triangle is defined as a triangle where all three angles are less than 90°.

Question 12:

A triangle has sides of lengths 5 cm, 12 cm, and 13 cm. Verify whether it is a right-angled triangle using the Pythagoras Theorem. Also, name the type of triangle based on its sides.

Answer:

To verify if the triangle is right-angled, we use the Pythagoras Theorem, which states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Given sides: 5 cm, 12 cm, 13 cm.

Step 1: Identify the longest side (hypotenuse). Here, it is 13 cm.

Step 2: Apply Pythagoras Theorem:
(5)² + (12)² = (13)²
25 + 144 = 169
169 = 169

Since both sides are equal, the triangle satisfies the Pythagoras Theorem. Hence, it is a right-angled triangle.

Type of triangle based on sides: Here, all three sides (5 cm, 12 cm, 13 cm) are of unequal lengths. Therefore, it is a scalene triangle as well.

Question 13:

In a triangle ABC, the measure of angle A is 50° and angle B is 70°. Find the measure of angle C. Also, classify the triangle based on its angles and sides. Justify your answer with proper reasoning.

Answer:

To find the measure of angle C, we use the Angle Sum Property of a triangle, which states that the sum of all interior angles of a triangle is 180°.

Given:
Angle A = 50°
Angle B = 70°

Step 1: Add angles A and B.
50° + 70° = 120°

Step 2: Subtract the sum from 180° to find angle C.
180° - 120° = 60°

Thus, angle C = 60°.

Now, let's classify the triangle:

Based on angles: Since all angles (50°, 70°, 60°) are less than 90°, it is an acute-angled triangle.

Based on sides: Since all angles are unequal, the sides opposite to them will also be unequal. Hence, it is a scalene triangle.

Question 14:

A triangle has sides measuring 5 cm, 12 cm, and 13 cm. Verify whether it is a right-angled triangle using the Pythagoras Theorem. Also, name the type of triangle based on its sides.

Answer:

Now, classify the triangle based on its sides:

Since all three sides (5 cm, 12 cm, 13 cm) are of different lengths, it is a scalene triangle.

Question 15:

In a triangle ABC, the measure of angle A is 50° and angle B is 70°. Find the measure of angle C. Also, classify the triangle based on its angles and sides.

Answer:

To find the measure of angle C in triangle ABC, we use the Angle Sum Property of a triangle, which states that the sum of all interior angles is 180°.

Given:
∠A = 50°
∠B = 70°

Step 1: Add ∠A and ∠B.
50° + 70° = 120°

Step 2: Subtract the sum from 180° to find ∠C.
180° - 120° = 60°

Thus, ∠C = 60°.

Now, let's classify the triangle:

Based on angles: Since all angles (50°, 70°, 60°) are less than 90°, it is an acute-angled triangle.
Based on sides: If all sides are unequal, it is a scalene triangle. If any two sides are equal, it would be isosceles. Without side lengths, we assume it is scalene unless stated otherwise.

Question 16:

A ladder 10 m long is placed against a wall such that its foot is 6 m away from the wall. Using the Pythagoras theorem, find the height at which the ladder touches the wall. Also, explain the practical application of this theorem in real life.

Answer:

Here, the ladder, wall, and ground form a right-angled triangle. The ladder acts as the hypotenuse, the distance from the wall (6 m) is one leg, and the height reached is the other leg.

Given:
Hypotenuse (ladder) = 10 m
Base (distance from wall) = 6 m
Height = ?

Step 1: Apply Pythagoras theorem (Hypotenuse² = Base² + Height²).
10² = 6² + Height²
100 = 36 + Height²

Step 2: Subtract 36 from both sides.
Height² = 100 - 36 = 64

Step 3: Take the square root.
Height = √64 = 8 m

Thus, the ladder touches the wall at 8 meters.

Practical Application: The Pythagoras theorem is used in construction to ensure right angles, in navigation to calculate shortest distances, and in designing ramps or staircases for safety and efficiency.

Question 17:

Explain the Pythagorean theorem with a suitable example. Also, state how this theorem is useful in real-life applications.

Answer:

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is written as: a² + b² = c², where c is the hypotenuse, and a and b are the other two sides.

Example: Consider a right-angled triangle with sides 3 cm and 4 cm. To find the hypotenuse:

Step 1: Apply the formula: a² + b² = c²
Step 2: Substitute the values: 3² + 4² = c²
Step 3: Calculate: 9 + 16 = c² → 25 = c²
Step 4: Take the square root: c = √25 → c = 5 cm

Real-life applications: The Pythagorean theorem is used in construction to ensure corners are square, in navigation to calculate distances, and in designing ramps or staircases. It helps architects and engineers create accurate and stable structures.

Question 18:

Define the exterior angle property of a triangle with a diagram. Using this property, find the measure of the exterior angle if the two opposite interior angles are 45° and 55°.

Answer:

The exterior angle property of a triangle states that the measure of an exterior angle is equal to the sum of the measures of the two opposite interior angles. In other words, for any triangle, the exterior angle at a vertex is the sum of the two non-adjacent interior angles.

Diagram: Imagine a triangle ABC with side BC extended to point D. The angle ACD is the exterior angle, and angles BAC and ABC are the opposite interior angles.

Given: Two opposite interior angles are 45° and 55°.

Step 1: Recall the property: Exterior angle = Sum of opposite interior angles
Step 2: Add the given angles: 45° + 55° = 100°
Step 3: Conclusion: The exterior angle measures 100°.

This property is useful in solving problems related to angles in polygons and verifying geometric constructions.

Case-based Questions (4 Marks) – with Solutions (CBSE Pattern)

These 4-mark case-based questions assess analytical skills through real-life scenarios. Answers must be based on the case study provided.

Question 1:

A ladder leans against a wall forming a right-angled triangle. The ladder is 5 m long, and its base is 3 m from the wall. Using the Pythagorean theorem, find the height where the ladder touches the wall.

Answer:

Problem Interpretation

We studied that a ladder, wall, and ground form a right-angled triangle. The ladder is the hypotenuse.

Mathematical Modeling

Using the Pythagorean theorem: Hypotenuse² = Base² + Height².

Solution

Given: Hypotenuse (ladder) = 5 m, Base = 3 m.
Height² = 5² - 3² = 25 - 9 = 16.
Height = √16 = 4 m.

Question 2:

In ΔABC, ∠A = 50° and ∠B = 70°. Using the angle sum property, find ∠C. Also, classify the triangle based on its angles.

Answer:

Problem Interpretation

Our textbook shows that the sum of angles in a triangle is 180°.

Mathematical Modeling

∠A + ∠B + ∠C = 180°.

Solution

Given: ∠A = 50°, ∠B = 70°.
∠C = 180° - (50° + 70°) = 60°.
Since all angles are less than 90°, it is an acute-angled triangle.

Question 3:

A ladder leans against a wall forming a right-angled triangle. The ladder is 5 m long, and its base is 3 m from the wall. Using the Pythagoras theorem, find the height the ladder reaches on the wall.

Answer:

Problem Interpretation

We studied that a ladder, wall, and ground form a right-angled triangle. The ladder is the hypotenuse.

Mathematical Modeling

Using Pythagoras theorem: Hypotenuse² = Base² + Height².

Solution

Given: Hypotenuse (ladder) = 5 m, Base = 3 m.
Height² = 5² - 3² = 25 - 9 = 16.
Height = √16 = 4 m.

The ladder reaches 4 m high on the wall.

Question 4:

In a right-angled triangle ABC, angle B is 90°. Side AB = 6 cm and BC = 8 cm.
(i) Find the length of AC using the Pythagoras theorem.
(ii) If another triangle PQR has sides PQ = 3 cm, QR = 4 cm, and PR = 5 cm, is it a right-angled triangle? Justify.

Answer:

Problem Interpretation

We studied that in a right-angled triangle, the Pythagoras theorem holds: hypotenuse² = base² + height².

Mathematical Modeling

For triangle ABC: AC² = AB² + BC² = 6² + 8² = 100 ⇒ AC = 10 cm.
For triangle PQR: Check if 5² = 3² + 4² ⇒ 25 = 9 + 16 ⇒ True. Hence, PQR is right-angled at Q.

Question 5:

A ladder 13 m long leans against a wall. Its foot is 5 m away from the wall.
(i) How high does the ladder reach on the wall?
(ii) If the foot is moved 1 m closer to the wall, what is the new height?

Answer:

Problem Interpretation

Our textbook shows real-life applications of the Pythagoras theorem for ladders and walls.

Mathematical Modeling

(i) Height² = 13² - 5² = 169 - 25 = 144 ⇒ Height = 12 m.
(ii) New base = 5 - 1 = 4 m ⇒ New height² = 13² - 4² = 169 - 16 = 153 ⇒ Height ≈ 12.37 m.

Question 6:

In ΔABC, ∠A = 50° and ∠B = 70°. Using the angle sum property of a triangle, find ∠C. Also, classify the triangle based on its angles.

Answer:

Problem Interpretation

Our textbook shows that the sum of angles in a triangle is 180°.

Mathematical Modeling

∠A + ∠B + ∠C = 180°.

Solution

Given: ∠A = 50°, ∠B = 70°
∠C = 180° - (50° + 70°) = 60°
Since all angles are less than 90°, it is an acute-angled triangle.

Question 7:

A ladder leans against a wall forming a right-angled triangle. The ladder is 5 m long, and its base is 3 m away from the wall. Using the Pythagoras theorem, find the height the ladder reaches on the wall.

Answer:

Problem Interpretation

We studied that a ladder, wall, and ground form a right-angled triangle. The ladder is the hypotenuse.

Mathematical Modeling

Using Pythagoras theorem: Hypotenuse² = Base² + Height².

Solution

Given: Hypotenuse = 5 m, Base = 3 m
Height² = 5² - 3² = 25 - 9 = 16
Height = √16 = 4 m

The ladder reaches 4 m high on the wall.

Question 8:

In ΔABC, ∠A = 60° and ∠B = 50°. Find ∠C using the angle sum property of a triangle. Also, classify the triangle based on its angles.

Answer:

Problem Interpretation

Our textbook shows that the sum of angles in a triangle is 180°.

Mathematical Modeling

∠A + ∠B + ∠C = 180°.

Solution

Given: ∠A = 60°, ∠B = 50°
∠C = 180° - (60° + 50°) = 70°
Since all angles are less than 90°, it is an acute-angled triangle.

∠C measures 70°, and ΔABC is acute-angled.

Question 9:

In ΔABC, ∠A = 50° and ∠B = 70°. Find ∠C using the angle sum property of a triangle. Also, classify the triangle based on its angles.

Answer:

Problem Interpretation

Our textbook shows that the sum of angles in a triangle is 180°.

Mathematical Modeling

∠A + ∠B + ∠C = 180°.

Solution

Given: ∠A = 50°, ∠B = 70°
∠C = 180° - (50° + 70°) = 60°
Since all angles are less than 90°, it is an acute-angled triangle.

Question 10:

A ladder leans against a wall forming a right-angled triangle. The ladder is 5 m long, and its base is 3 m from the wall. Using the Pythagorean theorem, find the height the ladder reaches on the wall.

Answer:

Problem Interpretation

We studied that in a right-angled triangle, the square of the hypotenuse equals the sum of squares of the other two sides.

Mathematical Modeling

Here, the ladder is the hypotenuse (5 m), and the base is one side (3 m). Let the height be 'h'.

Solution

Using the theorem: 5² = 3² + h² → 25 = 9 + h² → h² = 16 → h = 4 m.

Question 11:

A ladder leans against a wall forming a right-angled triangle. The base is 3m, and the ladder is 5m long. Find the height using the Pythagoras theorem.

Answer:

Problem Interpretation

We studied right-angled triangles in our textbook. Here, the ladder acts as the hypotenuse.

Mathematical Modeling

Using Pythagoras theorem: Hypotenuse² = Base² + Height².

Solution

Given: Base = 3m, Hypotenuse = 5m.
Height² = 5² - 3² = 25 - 9 = 16.
Height = √16 = 4m.

Question 12:

A triangular park has sides 6m, 8m, and 10m. Verify if it is a right-angled triangle using the Pythagoras property.

Answer:

Problem Interpretation

Our textbook shows that a triangle is right-angled if it satisfies Pythagoras theorem.

Mathematical Modeling

Check if 6² + 8² = 10².

Solution

6² + 8² = 36 + 64 = 100.
10² = 100.
Since both are equal, it is a right-angled triangle.

Question 13:

Riya has a triangular garden with sides measuring 5 m, 12 m, and 13 m. She wants to fence it. Help her determine:

Whether the garden is a right-angled triangle.
The total length of fencing required if an extra 2 m is needed for the gate.

Answer:

Step 1: Check if it's a right-angled triangle
Using the Pythagoras theorem, we verify if 5² + 12² = 13².
25 + 144 = 169
169 = 169 (True).
Thus, the garden is a right-angled triangle.

Step 2: Calculate fencing length
Perimeter = 5 m + 12 m + 13 m = 30 m.
Add extra 2 m for the gate: 30 m + 2 m = 32 m.

Note: Always verify sides using Pythagoras theorem for right-angled triangles.

Question 14:

A triangle has angles in the ratio 2:3:4. Find:

The measure of each angle.
Classify the triangle based on its angles.

Answer:

Step 1: Find angle measures
Let the angles be 2x, 3x, and 4x.
Sum of angles in a triangle = 180°.
2x + 3x + 4x = 180°
9x = 180°
x = 20°.
Angles are:
2x = 40°
3x = 60°
4x = 80°.

Step 2: Classify the triangle
Since all angles are less than 90°, it is an acute-angled triangle.

Tip: Ratios help distribute total angle sum proportionally.

Question 15:

In a park, three children are standing such that the distance between the first and second child is 5 meters, the second and third child is 12 meters, and the third and first child is 13 meters. The children wonder if they form a right-angled triangle. Verify their claim using the Pythagoras theorem and explain your reasoning.

Answer:

To verify if the children form a right-angled triangle, we can use the Pythagoras theorem, which states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Given distances:
Side 1 (a) = 5 meters
Side 2 (b) = 12 meters
Side 3 (c) = 13 meters (longest side, possible hypotenuse)

Step 1: Check if 13² = 5² + 12²
Step 2: Calculate 13² = 169
Step 3: Calculate 5² + 12² = 25 + 144 = 169
Step 4: Compare both results: 169 = 169

Since the condition of the Pythagoras theorem is satisfied, the children indeed form a right-angled triangle with the right angle between the sides of 5 meters and 12 meters.

Question 16:

A triangular garden has sides of lengths 7 meters, 24 meters, and 25 meters. The gardener wants to install a fence around it but is unsure if the garden is a right-angled triangle. Help him determine this using the Pythagoras theorem and explain whether the fence will fit perfectly.

Answer:

To determine if the garden is a right-angled triangle, we apply the Pythagoras theorem, which requires the square of the longest side to equal the sum of the squares of the other two sides.

Given sides:
Side 1 (a) = 7 meters
Side 2 (b) = 24 meters
Side 3 (c) = 25 meters (longest side, possible hypotenuse)

Step 1: Check if 25² = 7² + 24²
Step 2: Calculate 25² = 625
Step 3: Calculate 7² + 24² = 49 + 576 = 625
Step 4: Compare both results: 625 = 625

The garden satisfies the Pythagoras theorem, confirming it is a right-angled triangle. Since the sides form a perfect right-angled triangle, the fence will fit perfectly around the garden.

Question 17:

In a park, three children are standing at points A, B, and C forming a triangle. The distances between them are AB = 5 meters, BC = 7 meters, and AC = 10 meters. Using the triangle inequality property, determine if the points can form a triangle. Justify your answer with steps.

Answer:

To check if the points A, B, and C can form a triangle, we use the triangle inequality property, which states that the sum of any two sides of a triangle must be greater than the third side.

Let's verify all three conditions:
1. AB + BC > AC → 5 + 7 > 10 → 12 > 10 (True)
2. BC + AC > AB → 7 + 10 > 5 → 17 > 5 (True)
3. AB + AC > BC → 5 + 10 > 7 → 15 > 7 (True)

Since all conditions are satisfied, the points A, B, and C can form a valid triangle.

Question 18:

A ladder leaning against a wall forms a right-angled triangle with the wall and the ground. The ladder is 13 meters long, and its base is 5 meters away from the wall. Using the Pythagoras theorem, find the height at which the ladder touches the wall. Show your calculations.

Answer:

In this right-angled triangle, the ladder acts as the hypotenuse, the distance from the wall is one leg, and the height on the wall is the other leg.

Using the Pythagoras theorem:
Hypotenuse² = Base² + Height²
13² = 5² + Height²
169 = 25 + Height²
Height² = 169 - 25 = 144
Height = √144 = 12 meters

Thus, the ladder touches the wall at a height of 12 meters.

Question 19:

In a park, three children are standing such that the distance between the first and the second child is 5 meters, the second and the third child is 12 meters, and the third and the first child is 13 meters.

Check if the positions of the children form a right-angled triangle. Justify your answer using the Pythagoras theorem.

Answer:

To check if the positions form a right-angled triangle, we can use the Pythagoras theorem, which states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.

Given distances:
Side 1 (a) = 5 meters
Side 2 (b) = 12 meters
Side 3 (c) = 13 meters (longest side, possible hypotenuse)

Step 1: Calculate the sum of squares of the two shorter sides:
a² + b² = 5² + 12² = 25 + 144 = 169

Step 2: Calculate the square of the longest side:
c² = 13² = 169

Since a² + b² = c² (169 = 169), the Pythagoras theorem holds true.

Conclusion: The positions of the children form a right-angled triangle with the right angle between the sides of 5 meters and 12 meters.

Question 20:

A ladder 10 meters long is placed against a wall. The foot of the ladder is 6 meters away from the wall.

Determine the height at which the ladder touches the wall and identify the type of triangle formed by the ladder, wall, and ground.

Answer:

This scenario forms a right-angled triangle where:
- The ladder acts as the hypotenuse (10 meters).
- The distance from the foot of the ladder to the wall is one side (6 meters).
- The height at which the ladder touches the wall is the other side (to be calculated).

Using the Pythagoras theorem:
Hypotenuse² = Base² + Height²

Step 1: Rearrange the formula to find the height:
Height² = Hypotenuse² - Base²
Height² = 10² - 6² = 100 - 36 = 64

Step 2: Take the square root of the result:
Height = √64 = 8 meters

Conclusion: The ladder touches the wall at a height of 8 meters, and the triangle formed is a right-angled triangle with sides 6 meters, 8 meters, and 10 meters.

Question 21:

In a park, three children are standing at points A, B, and C forming a triangle. The distances between them are AB = 5 meters, BC = 12 meters, and AC = 13 meters.

(i) Identify the type of triangle formed by the children.
(ii) Justify your answer using the Pythagoras theorem.

Answer:

(i) The triangle formed is a right-angled triangle because one of its angles is 90°.

(ii) To justify using the Pythagoras theorem, we check if the sum of the squares of the two shorter sides equals the square of the longest side:
AB² + BC² = 5² + 12² = 25 + 144 = 169
AC² = 13² = 169
Since AB² + BC² = AC², the triangle satisfies the Pythagoras theorem, confirming it is right-angled at B.

Question 22:

A ladder 10 meters long leans against a wall. The foot of the ladder is 6 meters away from the wall.

(i) Draw a diagram representing the situation.
(ii) Find the height on the wall where the ladder touches it. Use the Pythagoras theorem.

Answer:

(i) Diagram:
[Representation: A right-angled triangle with the ladder as the hypotenuse (10 m), the distance from the wall as the base (6 m), and the height on the wall as the perpendicular.]

(ii) Using the Pythagoras theorem:
Let the height be h meters.
Hypotenuse² = Base² + Perpendicular²
10² = 6² + h²
100 = 36 + h²
h² = 100 - 36 = 64
h = √64 = 8 meters
The ladder touches the wall at a height of 8 meters.

The Triangle and Its Properties – CBSE NCERT Study Resources

Overview of the Chapter: The Triangle and Its Properties

1. Introduction to Triangles

2. Types of Triangles

3. Properties of Triangles

4. Pythagoras Theorem

5. Medians and Altitudes of a Triangle

6. Equilateral and Isosceles Triangles

7. Sum of the Lengths of Two Sides of a Triangle

8. Right-Angled Triangles and Pythagoras Property

All Question Types with Solutions – CBSE Exam Pattern

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