In ΔABC, D is a point on side BC such that ∠ADC = ∠BAC. If CD = 4 cm and CB = 9 cm, find CA.

Class 10Triangles (Similarity)Geometry — Similarity

A6 cm
B36 cm
C13 cm
D2.25 cm

Step-by-Step Solution

In Δ

A D C

and Δ

B A C

, ∠

C

is common and ∠

A D C

= ∠

B A C

, so Δ

A D C

~ Δ

B A C (AA)

. Matching the correspondence A ↔

B, D

↔ A,

C

↔

C

gives

\frac{C D}{C A} = \frac{C A}{CB}

, i.e.

C A^{2} = C D \times CB = 4 \times 9 = 36

, so

C A = 6

cm. 36 cm is the value of

C

A^{2}

with the square root forgotten; 13 cm is 4 + 9; 2.25 cm is

\frac{9}{4}

Answer: A) 6 cm

Key idea: Three similarity criteria

**AA (Angle-Angle)**: Two angles equal → similar. (Third angle is automatic by 180° angle sum.)

**SSS (Side-Side-Side proportionality)**: All three pairs of corresponding sides in same ratio → similar.

**SAS (Side-Angle-Side)**: Two pairs of sides proportional AND included angles equal → similar.

**Important**: SSA does NOT prove similarity (just like for congruence).

AA is by far the most-used in problems: spot two equal angles (often via parallel lines or shared angle) and declare similarity.

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