Chapter 11.7, Problem 21HP

To determine

To find: Vertices of Original figure.

Answer to Problem 21HP

Coordinates of the Original image are

  $\begin{array}{l}A\left(-2,-4\right)\\ B\left(-6,-10\right)\\ C\left(-10,-8\right)\\ D\left(-12,-4\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}k=\frac{1}{2}\\ A\text{'}=\left(1,2\right)\\ B\text{'}=\left(3,5\right)\\ C\text{'}=\left(5,4\right)\\ D\text{'}=\left(6,2\right)\end{array}$

The dilation is $\left(x,y\right)\to \left(\frac{1}{2}x,\frac{1}{2}y\right)$ .

Multiply the coordinates of each vertex by $\frac{1}{2}$ to get Dilated Vertices.

But when Dilated vertices are given, we can find Original vertices by

Original is $A\text{'}\left(x,y\right)\to A\left(\frac{1}{k}\times x,\frac{1}{k}\times y\right)$

Here $k=\frac{1}{2}$

  $\Rightarrow \frac{1}{k}=\frac{1}{\frac{1}{2}}=2$

It means we need to Multiply coordinates of each Dilated Vertex by $2$ to get Original Vertices.

  $\begin{array}{l}A\text{'}\left(1,2\right)\to A\left(2\left(1\right),2\left(2\right)\right)\\ \Rightarrow A\text{'}\left(1,2\right)\to A\left(2,4\right)\end{array}$

  $\begin{array}{l}B\text{'}\left(3,5\right)\to B\left(2\left(3\right),2\left(5\right)\right)\\ \Rightarrow B\text{'}\left(2,0\right)\to B\left(6,10\right)\end{array}$

  $\begin{array}{l}C\text{'}\left(5,4\right)\to C\left(2\left(5\right),2\left(4\right)\right)\\ \Rightarrow C\text{'}\left(5,4\right)\to C\left(10,8\right)\end{array}$

  $\begin{array}{l}D\text{'}\left(6,2\right)\to D\left(2\left(6\right),2\left(2\right)\right)\\ \Rightarrow D\text{'}\left(6,2\right)\to D\left(12,4\right)\end{array}$

Therefore,

Coordinates before reflections are

  $\begin{array}{l}A\left(2,4\right)\\ B\left(6,10\right)\\ C\left(10,8\right)\\ D\left(12,4\right)\end{array}$

Here as they did First reflection of Original coordinates with X-axis and then Y-axis to get Final coordinates.

Now we are doing in reverse process like we know final Coordinates so in reverse process we need to do reflection about Y-axis first and then reflection about X-axis and Dilation using scale factor.

We can also do dilation process before or after reflections

On reflecting about Y-axis

We get

  $\begin{array}{l}A\left(-2,4\right)\\ B\left(-6,10\right)\\ C\left(-10,8\right)\\ D\left(-12,4\right)\end{array}$

Again, reflecting about X-axis

We get

  $\begin{array}{l}A\left(-2,-4\right)\\ B\left(-6,-10\right)\\ C\left(-10,-8\right)\\ D\left(-12,-4\right)\end{array}$

Therefore,

Coordinates of the Original image are

  $\begin{array}{l}A\left(-2,-4\right)\\ B\left(-6,-10\right)\\ C\left(-10,-8\right)\\ D\left(-12,-4\right)\end{array}$

Graph of image is