Triangles

Board: CBSE | Class: Class 10

Comprehensive study notes for Triangles by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Similarity of Triangles

Two triangles are similar if their corresponding angles are equal and corresponding sides are proportional. Symbol: ΔABC ~ ΔDEF.

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides them in the same ratio. In ΔABC, if DE ∣ BC, then AD/DB = AE/EC.

Converse of BPT

If a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

Criteria for Similarity

AAA (or AA): If two angles are equal, triangles are similar. SSS: If all three sides are proportional. SAS: If two sides are proportional and the included angle is equal.

Areas of Similar Triangles

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If ΔABC ~ ΔDEF, then ar(ΔABC)/ar(ΔDEF) = (AB/DE)² = (BC/EF)² = (AC/DF)².

Pythagoras Theorem

In a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. AC² = AB² + BC². Its converse is also true.

Important Formulas

BPT / Thales	If DE ∣ BC, then AD/DB = AE/EC
Similarity (AAA)	If ∠A=∠D, ∠B=∠E, ∠C=∠F, then ΔABC ~ ΔDEF
Similarity (SSS)	If AB/DE = BC/EF = AC/DF, then ΔABC ~ ΔDEF
Areas of Similar	ar(ΔABC)/ar(ΔDEF) = (AB/DE)²
Pythagoras	AC² = AB² + BC² (for right Δ with ∠B = 90°)

Solved Examples

Example 1: In ΔABC, DE ∣ BC, AD = 3 cm, DB = 6 cm, AE = 4 cm. Find EC.

Solution: By BPT: AD/DB = AE/EC ⇒ 3/6 = 4/EC ⇒ EC = 8 cm.

Example 2: Check if ΔABC ~ ΔDEF: AB=3, BC=4, CA=5; DE=6, EF=8, FD=10.

Solution: 3/6 = 4/8 = 5/10 = 1/2. Sides proportional. Yes, they are similar (SSS).

Example 3: Find AC in right ΔABC with ∠B = 90°, AB = 3 cm, BC = 4 cm.

Solution: AC² = 3² + 4² = 9 + 16 = 25. AC = 5 cm.

Practice Questions

1. In ΔABC, DE ∣ BC, AD = 4 cm, DB = 8 cm, EC = 6 cm. Find AE.
2. A 15 m ladder reaches 9 m up a wall. How far is the foot from the wall?
3. Prove that the ratio of areas of two similar triangles equals the square of their side ratio.
4. In ΔABC, DE ∣ BC and AD:DB = 2:3. If AE = 4 cm, find EC.
5. Check if a triangle with sides 5, 12, 13 is right-angled.

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