EQN-04-07

Similar & Congruent Triangles

Use proportional sides in similar triangles to find missing lengths — a high-frequency GAT-E figure type.

~6 min read•Difficulty 3/5

Congruent: same shape AND same size (all sides and angles equal).
Similar: same shape, possibly different size — equal angles, and sides in a constant ratio (the scale factor).

In similar triangles, corresponding sides are proportional.

small side / big side = small side / big side

Match corresponding sides carefully (shortest to shortest, etc.) and cross-multiply.

Parallel lines create equal corresponding angles $\to$ similar triangles (common in "triangle inside a triangle" figures).
A line parallel to one side of a triangle cuts the other two sides proportionally.

If the scale factor is k, sides scale by k, areas by k². Two similar triangles with sides in ratio 2 : 3 have areas in ratio 4 : 9.

Diagram

Similar triangles have equal angles and sides in proportion (here 6 : 9).

Worked examples

Example 1 (difficulty 2/5)

Two similar triangles have corresponding sides 6 and 9. A side of the small triangle is 8. What is the matching side of the large one?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Two similar triangles have sides in ratio 3 : 5. What is the ratio of their areas?

◎ Predict the answer first:

Pairing sides that do not correspond

Why: The figure may be rotated or flipped.

Fix: Match by position (shortest↔shortest, opposite equal angles) before setting up the ratio.

Scaling area by the side ratio instead of its square

Why: Area feels like another length.

Fix: Area ratio

= (

side ratio)².