Chapter 3.6, Problem 20WE

To determine

Whether the given statement is true or false. Justify with a diagram.

Answer to Problem 20WE

A triangle have equal perimeters, then they have congruent sides is false.

Explanation of Solution

Given information:

Given statement:

Two triangles have equal perimeters, then they have congruent sides..

Calculation:

Consider the following figure:

In above figure there is one equilateral $\Delta ABC$ and a right $\Delta PQR$

In which it mention that

  $\text{Perimeter}\text{of}\Delta ABC=\text{Perimeter}\text{of}\Delta PQR$

To prove $AB\cong PQ,BC\cong QR$ and $AC\cong PR$

The Perimeter of equilateral triangle $=3a$ , where $a=4$

  $\begin{array}{c}\text{Perimeter}\text{of}\Delta ABC=3\left(4\right)\\ =12\end{array}$

$\text{Perimeter}\text{of}\text{right}\text{triangle}=a+b+c,\text{where}a=3,b=4$ and $c=5$

  $\begin{array}{c}\text{Perimeter}\text{of}\Delta PQR=3+4+5\\ =12\end{array}$

So according to above definitions

  $\text{Perimeter}\text{of}\Delta ABC=\text{Perimeter}\text{of}\Delta PQR$

But sides $BC\ne QR$ and $AC\ne PR$

Therefore, if a triangle have equal perimeters, then they have congruent sides is false.