Chapter 6.1, Problem 43E

To determine

To calculate : the ratio of the area of the shaded region to the area of the unshaded region.

Answer to Problem 43E

The ratio of the area of the shaded region to the area of the unshaded region is $1:1$ .

Explanation of Solution

Given information :

Length of each side of square is given as $s$ .

The shaded region is a tringle having base and height equal to $s$ .

Formula used :

Formula used for area of square,

  $\text{Area}={\left(\text{side}\right)}^2$

Formula used for area of triangle,

  $\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}$

Calculation :

The ratio of the area of the shaded region to the area of the unshaded region can be derived as,

Area of unshaded region = area of square − area of shaded region.

The area of square,

  $\begin{array}{l}\text{Area}={\left(\text{side}\right)}^2\\ \text{Area}={\left(s\right)}^2\\ \text{Area}=s^2\end{array}$

The area of the shaded region or triangle,

  $\begin{array}{l}\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}\\ \text{Area}=\frac{1}{2}\times s\times s\\ \text{Area}=\frac{1}{2}{s}^2\end{array}$

Area of unshaded region = area of square − area of shaded region

Area of unshaded region = $s^2-\frac{1}{2}{s}^2$

Area of unshaded region = $\frac{1}{2}{s}^2$

The ratio of the area of the shaded region to the area of the unshaded region can be derived as,

The ratio of area of shaded region : ratio of area of unshaded region

  $\begin{array}{l}=\frac{1}{2}{s}^2:\frac{1}{2}{s}^2\\ =1:1\end{array}$

Therefore,

The ratio of the area of the shaded region to the area of the unshaded region is $1:1$ .