Chapter 6 · Question 5

What is the theorem on areas of similar triangles?

Answer

Direct Answer

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

\triangle ABC\sim\triangle PQR

, then

\frac{\text{ar}(ABC)}{\text{ar}(PQR)}=\left(\frac{AB}{PQ}\right)^2

.

Simple Explanation

For similar triangles, area ratio is side ratio squared.

Exam-Ready Structure

If two triangles are similar, all corresponding lengths are in the same ratio. Since area depends on two dimensions, the area ratio becomes the square of the side ratio. Thus

\frac{\text{ar}(ABC)}{\text{ar}(PQR)}=\frac{AB^2}{PQ^2}=\frac{BC^2}{QR^2}=\frac{CA^2}{RP^2}

.

Key Points

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