Chapter 7.3, Problem 3WE

To determine

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

Answer to Problem 3WE

Two isosceles triangles are sometimes similar.

Explanation of Solution

Given information: Polygons are two isosceles 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.

A triangle in which two sides are equal is called as isosceles triangle.

Case 1:

Consider the following figures of two isosceles triangles:

  

Here, triangles ABC and PQR are isosceles as two sides in each of the triangles are equal.

  $\left\{AB=AC,PQ=PR\right\}$

Check whether the sides of triangles ABC and PQR are proportional.

  $\frac{AB}{PQ}=\frac{5}{6},\frac{AC}{PR}=\frac{5}{6},\frac{BC}{QR}=\frac{6}{8}=\frac{3}{4}$

So,

  $\frac{AB}{PQ}\ne \frac{BC}{QR}$

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

Case 2:

Consider the following figures:

  

Here, triangles ABC and PQR are isosceles as two sides in each of the triangles are equal.

  $\left\{AB=AC,PQ=PR\right\}$

  $\begin{array}{l}\angle A=\angle P=70°\\ \angle B=\angle Q=55°\\ \angle C=\angle R=55°\end{array}$

And

  $\begin{array}{l}\frac{AB}{PQ}=\frac{5}{10}=\frac{1}{2}\\ \frac{AC}{PR}=\frac{5}{10}=\frac{1}{2}\\ \frac{BC}{QR}=\frac{6}{12}=\frac{1}{2}\\ \therefore \frac{AB}{PQ}=\frac{AC}{PR}=\frac{BC}{QR}\end{array}$

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

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