Chapter 12.4, Problem 5MRE

a.

To determine

To find : Area of both trapezoids.

Answer to Problem 5MRE

The area of trapezoid ABCD and PQRS are 48 square units and 108 square units respectively.

Explanation of Solution

Given information :

The given trapezoids are

Calculation :

Since, the area of the trapezoid is given by

  $A=\frac{1}{2}\times \left(\text{Sum of the parallel sides}\right)\times \text{height}$

Therefore,

Area of the trapezoid ABCD is,

  $\begin{array}{l}A\left(ABCD\right)=\frac{1}{2}\times \left(12+4\right)\times \text{6}\\ A\left(ABCD\right)=16\times \text{3}\\ A\left(ABCD\right)=48\text{ square units}\end{array}$

Now, area of the trapezoid PQRS is,

  $\begin{array}{l}A\left(PQRS\right)=\frac{1}{2}\times \left(18+6\right)\times \text{9}\\ A\left(PQRS\right)=12\times \text{9}\\ A\left(PQRS\right)=108\text{ square units}\end{array}$

Hence,

The area of trapezoid ABCD and PQRS are 48 square units and 108 square units respectively.

b.

To determine

To find : The ratio of the area.

Answer to Problem 5MRE

The ratio is $9:4$.

Explanation of Solution

Given information :

The area of trapezoid ABCD and PQRS are 48 square units and 108 square units respectively.

Calculation :

From part (a) the area of trapezoid ABCD is 48 square units and PQRS is 108 square units.

Therefore, their ratios are

  $\begin{array}{l}\frac{A\left(PQRS\right)}{A\left(ABCD\right)}=\frac{108}{48}\\ \frac{PQRS}{ABCD}=\frac{9}{4}\\ \frac{PQRS}{ABCD}=9:4\end{array}$

Hence,

The ratio is $9:4$.

c.

To determine

To compare : the ratio of the area to the scale factor.

Answer to Problem 5MRE

The ratio of the area is equal to the square of the scale factor.

Explanation of Solution

Given information :

The ratio of the area is 9 : 4 and scale factor is 1.5.

Calculation :

The ratio of area is 9 : 4 and scale factor is 1.5.

Since, $\frac{9}{4}$ is not equivalent to 1.5.

Also,

  $\begin{array}{l}\frac{9}{4}={\left(\frac{3}{2}\right)}^2\\ \frac{9}{4}={\left(1.5\right)}^2\end{array}$

Hence,

The ratio of the area is equal to the square of the scale factor.