Chapter 7.3, Problem 2WE

To determine

To check: Whether two right triangles are always, sometimes or never similar.

Answer to Problem 2WE

Two right triangles are sometimes similar.

Explanation of Solution

Given information: Polygons are two right triangles

Two polygons are said to be similar if the following conditions are satisfied:

1. Corresponding sides of the polygons are proportional.
2. Corresponding angles of the polygons are equal.

Case 1:

Consider the following figures of two right triangles:

  

Here,

  $\begin{array}{l}\angle A=\angle P=30°\\ \angle C=\angle R=60°\\ \angle B=\angle Q=90°\end{array}$

So, corresponding angles are equal.

Now check whether the sides are proportional.

  $\begin{array}{l}\frac{AB}{PQ}=\frac{4}{12}=\frac{1}{3}\\ \frac{BC}{QR}=\frac{3}{5}\\ \frac{AC}{PR}=\frac{5}{13}\\ \therefore \frac{AB}{PQ}\ne \frac{BC}{QR}\ne \frac{AC}{PR}\end{array}$

In this case although corresponding angles are equal but as the corresponding sides are not proportional, triangles ABC and PQR are not similar.

Case 2:

Consider the following figures:

  

Here,

  $\begin{array}{l}\angle A=\angle P=30°\\ \angle C=\angle R=60°\\ \angle B=\angle Q=90°\end{array}$

So, corresponding angles are equal.

Now check whether the sides are proportional.

  $\begin{array}{l}\frac{AB}{PQ}=\frac{4}{12}=\frac{1}{3}\\ \frac{BC}{QR}=\frac{3}{9}=\frac{1}{3}\\ \frac{AC}{PR}=\frac{5}{15}=\frac{1}{3}\\ \therefore \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}\end{array}$

In this case as the corresponding angles are equal and the corresponding sides areproportional, triangles ABC and PQR aresimilar.

From cases 1 and 2, it can be observed that two right triangles may or not be similar that is two right triangles are sometimes similar.