Similarity of Triangles

ICSE Class 10 Mathematics • Chapter 10

1. Similar vs Congruent

Congruent (≅)	Similar (~)
Same shape AND size	Same shape, different size
All corresponding sides equal	All corresponding sides proportional
All corresponding angles equal	All corresponding angles equal
Symbol: △ABC ≅ △DEF	Symbol: △ABC ~ △DEF

2. Criteria for Similarity

Criterion	Full Form	Meaning
AA	Angle-Angle	Two pairs of corresponding angles are equal
SSS	Side-Side-Side	Three pairs of corresponding sides are proportional
SAS	Side-Angle-Side	Two pairs of sides proportional, included angle equal

AA is most commonly used! If two angles of one triangle equal two angles of another, the triangles are similar (third angle automatically equal).

3. Basic Proportionality Theorem (BPT / Thales' Theorem)

Statement: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

If DE ∥ BC in △ABC, then:

$\frac{AD}{DB} = \frac{AE}{EC}$

[Diagram: Triangle ABC with DE parallel to BC, cutting AB at D and AC at E]

Corollaries of BPT:

$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$
$\frac{DB}{AB} = \frac{EC}{AC}$

Converse of BPT

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

If $\frac{AD}{DB} = \frac{AE}{EC}$, then DE ∥ BC

4. Properties of Similar Triangles

If △ABC ~ △DEF with scale factor k (i.e., $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$), then:

Property	Ratio
Ratio of Sides	k
Ratio of Perimeters	k
Ratio of Altitudes	k
Ratio of Medians	k
Ratio of Areas	k²

Most Important: $\frac{\text{Area of } △ABC}{\text{Area of } △DEF} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{BC}{EF}\right)^2 = k^2$

5. Solved Examples

Example 1: In △ABC, DE ∥ BC. If AD = 4 cm, DB = 6 cm, AE = 5 cm, find EC.

By BPT: $\frac{AD}{DB} = \frac{AE}{EC}$

$\frac{4}{6} = \frac{5}{EC}$

$EC = \frac{6 \times 5}{4} = 7.5$ cm

Example 2: Two similar triangles have areas 36 cm² and 100 cm². If a side of smaller triangle is 12 cm, find corresponding side of larger triangle.

$\frac{Area_1}{Area_2} = \left(\frac{side_1}{side_2}\right)^2$

$\frac{36}{100} = \left(\frac{12}{x}\right)^2$

$\frac{6}{10} = \frac{12}{x}$

$x = 20$ cm

6. Applications: Scale and Maps

Scale: Ratio of map/model distance to actual distance.

If scale is 1:n, then:

Model length = $\frac{Actual length}{n}$
Model area = $\frac{Actual area}{n^2}$
Model volume = $\frac{Actual volume}{n^3}$

7. Quick Reference

Concept	Formula
BPT	$\frac{AD}{DB} = \frac{AE}{EC}$ (if DE ∥ BC)
Similar △ sides	Corresponding sides proportional
Area ratio	$(side ratio)^2$
Scale 1:n (Area)	Model area = $\frac{Actual}{n^2}$

Exam Practice (PYQ Trends)

PYQ: 2023

BOARD In △ABC, D and E are points on AB and AC such that DE ∥ BC. If $\frac{AD}{AB} = \frac{3}{5}$ and AC = 15 cm, find AE.

PYQ: 2022

BOARD Two similar triangles have areas in ratio 25:36. Find the ratio of corresponding altitudes.