Chapter 15.3, Problem 41E

To determine

Find the perimeter of the rectangleusing given data.

Answer to Problem 41E

The perimeter of the rectangle is $22$ inches.

Explanation of Solution

Given:

The givenstatement is that the area of the rectangle is equal to the area of the triangle. The sides of the triangle are $6,8$ , and $10$ inches and the width of the rectangle is $3$ inches.

Calculation:

For right angle triangle $c^2=a^2+b^2$ (Pythagoras theorem), where

  $c\to$ hypotenuse

  $a\to$ perpendicular

  $b\to$ base

In the given triangle

  ${10}^2=6^2+8^2$

So, the triangle is an right angle triangle.

Area of the right angle triangle $=\frac{1}{2}\times \text{base}\times \text{height}$

  $\begin{array}{l}=\frac{1}{2}\times \text{6}\times \text{8}\\ =24\text{square}\text{inches}\end{array}$

Now

Area of the rectangle $=$ Area of the right angle triangle $=24\text{square}\text{inches}$

Area of the rectangle $=\text{length}\times \text{width}$

Let

  $\text{length}=l$

  $\begin{array}{l}l\times \text{3=24}\\ \Rightarrow \frac{3l}{3}=\frac{24}{3}\\ \Rightarrow l=8\text{inches}\end{array}$

Perimeter of the rectangle $=2\left(\text{length}+\text{width}\right)$ $=2\left(\text{8}+\text{3}\right)=22$ inches.

Hence the perimeter of the rectangle is $22$ inches.