Willer Academy - Triangles

Willer Academy

Nation Builders Through Education | Founded by Er. Rahul Kumar Dubey Sir

Chapter 6: Triangles

Subject: Mathematics

Class: 10

Series: Base Builder for Moderate to Hard

Learning Objectives

By the end of this chapter, you, the future mathematician and nation-builder, will be able to:

Understand the properties and types of triangles
Apply various similarity criteria to prove triangle similarity
Use the Basic Proportionality Theorem (Thales Theorem) to solve problems
Apply Pythagoras Theorem and its converse to solve problems
Understand and apply area relationships in similar triangles
Solve problems involving triangles in real-life situations
Prove geometric theorems related to triangles

6.1 Introduction: Understanding Triangles

A triangle is a closed figure with three sides, three angles, and three vertices. It is one of the basic shapes in geometry and has numerous properties and applications.

Quick Check

Look around you and identify five objects that have triangular shapes. What purpose does the triangular shape serve in each object?

6.2 Types of Triangles

Triangles can be classified based on their sides or angles, each with unique properties.

By Sides

Scalene: All sides different
Isosceles: Two sides equal
Equilateral: All sides equal

By Angles

Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°

Similar Triangles

Same shape but different size
Corresponding angles equal
Corresponding sides proportional

Numerical & Application 1: Angle Sum Property

Scenario: In a triangle, two angles measure 45° and 60°.
Problem: What is the measure of the third angle? What type of triangle is it based on angles?
Solution:
- Sum of angles in a triangle = 180°
- Third angle = 180° - (45° + 60°) = 75°
- All angles are less than 90°, so it's an acute triangle

This demonstrates the angle sum property of triangles!

6.3 Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional.

Criterion	Explanation	Symbol
AAA	If corresponding angles are equal	∠A = ∠P, ∠B = ∠Q, ∠C = ∠R
SSS	If corresponding sides are proportional	AB/PQ = BC/QR = AC/PR
SAS	If one angle equal and including sides proportional	∠A = ∠P and AB/PQ = AC/PR

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

In ΔABC, if DE ∥ BC, then AD/DB = AE/EC

Task 1: Similarity Proof

In triangle ABC, points D and E are on sides AB and AC respectively such that DE ∥ BC. If AD = 3 cm, DB = 2 cm, and AE = 4.5 cm, find EC.

6.4 Pythagoras Theorem

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

Pythagoras Theorem

In a right triangle ABC, right-angled at B: AC² = AB² + BC²

Numerical & Application 2: Pythagoras Theorem

Scenario: A ladder 5 m long is placed against a wall such that it reaches a height of 4 m.
Problem: How far is the foot of the ladder from the wall?
Solution:
- Let the distance be x meters
- Using Pythagoras theorem: x² + 4² = 5²
- x² + 16 = 25 → x² = 9 → x = 3 m

This shows how Pythagoras theorem applies to real-world situations!

6.5 Area of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area Ratio

If ΔABC ~ ΔPQR, then
Area(ΔABC)/Area(ΔPQR) = (AB/PQ)² = (BC/QR)² = (AC/PR)²

Right Triangle Area

Area = ½ × base × height

Heron's Formula

Area = √[s(s-a)(s-b)(s-c)]
where s = (a+b+c)/2

Task 2: Area Calculation

Two similar triangles have corresponding sides in the ratio 3:5. What is the ratio of their areas? If the area of the smaller triangle is 36 cm², what is the area of the larger triangle?

6.6 Geometric Tools and Constructions

Geometric constructions help us understand triangle properties and relationships better.

Compass

Used for drawing arcs and circles, transferring distances

Ruler

Used for drawing straight lines and measuring lengths

Protractor

Used for measuring and constructing angles

Construction: Similar Triangles

Construct a triangle similar to a given triangle with scale factor 3/4. List the steps of your construction and verify that the constructed triangle is indeed similar to the original triangle.

Section B: Multiple Choice Questions

Q1. In two similar triangles, the ratio of their corresponding sides is 4:9. What is the ratio of their areas?

A. 2:3
B. 4:9
C. 16:81
D. 64:729

Q2. In a right triangle, the hypotenuse is 10 cm and one side is 6 cm. What is the length of the third side?

A. 4 cm
B. 6 cm
C. 8 cm
D. 12 cm

Q3. Which of the following is not a criterion for similarity of triangles?

A. AAA
B. SSS
C. SAS
D. SSA

Q4. In an equilateral triangle, each exterior angle measures:

A. 60°
B. 90°
C. 120°
D. 180°

Q5. The Basic Proportionality Theorem is also known as:

A. Pythagoras Theorem
B. Thales Theorem
C. Euclid's Theorem
D. Archimedes' Principle

Section C: Answer in Detail (For Answer Writing Practice)

📝 Multiple Choice Questions (MCQs)

In a right triangle, the square of the hypotenuse is equal to:
1. Sum of the squares of the other two sides
2. Difference of the squares of the other two sides
3. Product of the other two sides
4. None of these
If two triangles are similar, then their corresponding angles are:
1. Equal
2. Proportional
3. Right
4. Congruent
Which of the following is not a criterion for triangle similarity?
1. SSS
2. SAS
3. AAA
4. ASA
In ΔABC, DE || BC. If AD = 2 cm, DB = 4 cm, then AE : EC =
1. 1 : 2
2. 2 : 1
3. 1 : 3
4. 3 : 1
The sides of two similar triangles are in the ratio 2:3. What is the ratio of their areas?
1. 2 : 3
2. 4 : 9
3. 3 : 2
4. 9 : 4
The Pythagoras theorem is applicable only in:
1. Equilateral triangle
2. Right triangle
3. Isosceles triangle
4. Scalene triangle
In ΔXYZ ~ ΔPQR, XY = 6 cm, YZ = 8 cm, PQ = 9 cm. What is QR?
1. 10 cm
2. 12 cm
3. 15 cm
4. 8 cm
A triangle whose all sides are equal is called:
1. Scalene
2. Isosceles
3. Equilateral
4. Right-angled
In ΔABC, if AB = AC, then ΔABC is:
1. Scalene
2. Equilateral
3. Right-angled
4. Isosceles
Which tool is most commonly used to draw a circle?
1. Protractor
2. Compass
3. Set-square
4. Divider
The ratio of the areas of two similar triangles is equal to the square of the ratio of their:
1. Perimeters
2. Altitudes
3. Sides
4. Angles
A triangle with no equal sides is called:
1. Scalene
2. Isosceles
3. Equilateral
4. Right-angled
The Pythagoras theorem can also be written as:
1. a² + b² = c²
2. a² - b² = c²
3. a² × b² = c²
4. a² ÷ b² = c²
The square root of the area ratio of two similar triangles gives the ratio of their:
1. Sides
2. Angles
3. Perimeters
4. Altitudes
Which of the following is not a type of triangle?
1. Scalene
2. Isosceles
3. Equiangular
4. Square
The altitude drawn to the hypotenuse of a right triangle divides it into:
1. Two equal triangles
2. Two isosceles triangles
3. Two similar triangles
4. None of these
Which similarity criterion is used when two triangles have one equal angle and the sides including these angles are proportional?
1. AAA
2. SSS
3. SAS
4. ASA
The line drawn from the vertex of a triangle perpendicular to the opposite side is called:
1. Median
2. Altitude
3. Angle bisector
4. Perpendicular bisector
The longest side of a right-angled triangle is called:
1. Base
2. Altitude
3. Hypotenuse
4. Median
In ΔABC, if ∠A = 90°, AB = 3 cm, AC = 4 cm, then BC =
1. 5 cm
2. 6 cm
3. 7 cm
4. 12 cm

✅ Answer Key (MCQs)

Q1. State and prove the Basic Proportionality Theorem. Use a diagram to illustrate the theorem.

Hint: Include given, to prove, construction, and proof with proper reasoning.

Q2. Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Hint: Use the formula for area of triangle and property of similar triangles.

Q3. Explain the converse of Pythagoras Theorem with an example. How is it used to determine if a triangle is right-angled?

Hint: Include statement, explanation, and a numerical example.

Section D: Tackle These! (Higher Order Thinking Skills - HOTS)

Q1. In an equilateral triangle ABC, D is a point on side BC such that BD = 1/3 BC. Prove that 9AD² = 7AB².

Think about: Using Pythagoras theorem in right triangles formed by drawing altitude.

Q2. The perpendicular from A on side BC of a ΔABC intersects BC at D such that DB = 3CD. Prove that 2AB² = 2AC² + BC².

Think about: Applying Pythagoras theorem in triangles ABD and ACD.

Q3. In a right triangle, the hypotenuse is 5 cm. If one side is increased by 2 cm and the other side is increased by 3 cm, the hypotenuse becomes 6√2 cm. Find the lengths of the sides of the original triangle.

Think about: Setting up equations using Pythagoras theorem.