Chapter 7.3, Problem 4WE

To determine

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

Answer to Problem 4WE

Two scalene triangles are sometimes similar.

Explanation of Solution

Given information: Polygons are two scalene 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 all three sides are of different lengths and all angles are of different measures is known as a scalene triangle.

Case 1:

Consider the following figures of two isosceles triangles:

  

Here, triangles ABC and PQR are scalene as all sides in each of the triangles are of different lengths.

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

  $\begin{array}{l}\frac{AB}{PQ}=\frac{5}{8},\frac{AC}{PR}=\frac{8}{15},\frac{BC}{QR}=\frac{7}{11}\\ \therefore \frac{AB}{PQ}\ne \frac{AC}{PR}\ne \frac{BC}{QR}\end{array}$

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 scaleneas the sides are of different lengths and angles are of different measures.

Now check whether the corresponding angles of triangles ABC and PQR are equal or not.

  $\begin{array}{l}\angle A=\angle P=85°\\ \angle B=\angle Q=50°\\ \angle C=\angle R=45°\end{array}$

And

  $\begin{array}{l}\frac{AB}{PQ}=\frac{5}{10}=\frac{1}{2}\\ \frac{AC}{PR}=\frac{8}{16}=\frac{1}{2}\\ \frac{BC}{QR}=\frac{7}{14}=\frac{1}{2}\\ \therefore \frac{AB}{PQ}=\frac{AC}{PR}=\frac{BC}{QR}\end{array}$

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 scalene triangles may or not be similar that is two scalene triangles are sometimes similar.