Chapter 6.8, Problem 20HP

To determine

To graph: The original figure ABCD, stating its vertices on applying said transformation, so that to obtain given imaged figure.

Answer to Problem 20HP

Coordinates of original figure ABCD be:

  $A\left(1,2\right),B\left(3,5\right),\text{C}\left(5,4\right),D(6,2)$

and its graph is shown as :

  

Explanation of Solution

Given information:

 A figure A`B`C`D` with its vertices $A`\left(1,2\right),B`\left(3,5\right),\text{C`}\left(5,4\right),D`(6,2)$

That is obtained on applying following transformations on its original figure ABCD,

(i) Dilation by scale $k=\frac{1}{2}$(ii) Reflection over x and y axis.

Concept used:

By rule, in dilation, each original x and y coordinate is multiplied by the given factor and in reflection over x axis, the original coordinate $(x,y)$ is changed as,

  $\left(x,y\right)\to \left(x,-y\right)$

And in reflection over y axis, the original coordinates are changed as:

  $\left(x,y\right)\to \left(-x,y\right)$

If the original coordinate of vertex A be (x,y), then on applying all given transformations, the imaged coordinate will be $\left(-\frac{1}{2}x,-\frac{1}{2}y\right)$ .

Or if imaged coordinate is (x,y),then its original coordinate will be given as $\left(-2x,-2y\right)$ .

Calculation:

Based on above concept, $A=\left(-2\times 1,-2\times 2\right)=(-2,-4)$

  $\text{B}=\left(-2\times 3,-2\times 5\right)=\left(-6,-10\right)$

  $\text{C}=\left(-2\times 5,-2\times 4\right)=\left(-10,-8\right)$

  $\text{D}=\left(-2\times 6,-2\times 2\right)=\left(-12,-4\right)$

So, new corresponding vertices are: $A\left(1,2\right),B\left(3,5\right),\text{C}\left(5,4\right),D(6,2)$ .

Its graph is shown as: