Chapter 8.5, Problem 29IP

To determine

The lengths of the sides of the rectangle and of the triangle

Answer to Problem 29IP

The sides of rectangle is

  $\begin{array}{c}w=55\\ w+40=55+40\\ =95\end{array}$

The side of triangle is

  $\begin{array}{c}w+45=55+45\\ =100\end{array}$

Explanation of Solution

Given:

The perimeters of the two sections are equal and w represents the width of the rectangle

  

Concept Used:

Perimeter of rectangle $P=2\left(length+width\right)$

Perimeter of Triangle $P=Side+Side+Side$

Rules of Addition/ Subtraction:

• Two numbers with similar sign always get added and the resulting number will carry the similar sign.
• Two numbers with opposite signs always get subtracted and the resulting number will carry the sign of larger number.

Rules of Multiplication/ Division:

• The product/quotient of two similar sign numbers is always positive.
• The product/quotient of two numbers with opposite signs is always negative.

Calculation:

In order to find the perimeters of the rectangle $P=2\left(l+w\right)$ as below:

  $\begin{array}{c}P=2\left(length+width\right)\\ length=w\\ width=w+40\\ P=2\left(w+w+40\right)\\ =2\left(2w+40\right)\\ =2\cdot 2w+2\cdot 40\\ p=4w+80\end{array}$

In order to find the perimeters of the triangle $P=Side+Side+Side$ as below:

  $\begin{array}{c}P=Side+Side+Side\\ P=\left(w+45\right)+\left(w+45\right)+\left(w+45\right)\\ P=3w+135\end{array}$

Both are equal, thus the equation is $4w+80=3w+135$

In order to solve the equation $4w+80=3w+135$ first subtract both sides by $3w$ and then subtract both sides by $80$ and then simplify further as below:

  $\begin{array}{c}4w+80=3w+135\\ 4w+80-3w=3w+135-3w\\ w+80=135\\ w+80-80=135-80\\ w=55\end{array}$

Thus, the sides of rectangle is

  $\begin{array}{c}w=55\\ w+40=55+40\\ =95\end{array}$

The side of triangle is

  $\begin{array}{c}w+45=55+45\\ =100\end{array}$