Chapter 4.3, Problem 87STP

To determine

To calculate: The area of quadrilateral formed by $A\text{'}B\text{'}C\text{'}D\text{'}$ if $ABCD$ is transformed as $\left(x,y\right)\to \left(3x,4y\right)$ .

  

Answer to Problem 87STP

The area of quadrilateral formed by $A\text{'}B\text{'}C\text{'}D\text{'}$ is $192\text{ sq units}$ .

Explanation of Solution

Given information:

The $ABCD$ is transformed as $\left(x,y\right)\to \left(3x,4y\right)$ .

  

Formula used:

The area of rectangle whose length is l and width is w is $A=lw$ .

Calculation:

Consider the $ABCD$ is transformed as $\left(x,y\right)\to \left(3x,4y\right)$ .

  

The coordinates of $A\text{'}B\text{'}C\text{'}D\text{'}$ are.

Coordinate A is transformed as,

  $\begin{array}{c}\left(0,0\right)\to \left(3\left(0\right),4\left(0\right)\right)\\ \left(0,0\right)\to \left(0,0\right)\end{array}$

Coordinate B is transformed as,

  $\begin{array}{c}\left(4,0\right)\to \left(3\left(4\right),4\left(0\right)\right)\\ \left(4,0\right)\to \left(12,0\right)\end{array}$

Coordinate C is transformed as,

  $\begin{array}{c}\left(4,4\right)\to \left(3\left(4\right),4\left(4\right)\right)\\ \left(4,4\right)\to \left(12,16\right)\end{array}$

Coordinate D is transformed as,

  $\begin{array}{c}\left(0,4\right)\to \left(3\left(0\right),4\left(4\right)\right)\\ \left(0,4\right)\to \left(0,16\right)\end{array}$

Therefore new coordinates are $A\text{'}=\left(0,0\right),B\text{'}=\left(12,0\right),C\text{'}=\left(12,16\right)\text{ and }D\text{'}=\left(0,16\right)$ .

  

This forms are rectangle. Apply the area of rectangle whose length is l and width is w is $A=lw$ .

Here length is 12 and width is 16.

  $\begin{array}{c}A=12\times 16\\ =192\end{array}$

Thus, the area of quadrilateral formed by $A\text{'}B\text{'}C\text{'}D\text{'}$ is $192\text{ sq units}$ .