Chapter 11.7, Problem 22HP

a.

To determine

To find: Vertices of each figure after a dilation.

Answer to Problem 22HP

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,4\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(8,4\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}A=\left(0,5\right)\\ B=\left(0,0\right)\\ C=\left(10,5\right)\end{array}$

  $k=2$ followed by $k=\frac{2}{5}$

First dilation is $\left(x,y\right)\to \left(2x,2y\right)$ .

Multiply the coordinates of each vertex by $2$ .

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

Again, on dilation with scale factor $k=\frac{2}{5}$

We get

  $\begin{array}{l}A\text{'}\left(0,10\right)\to A\text{'}\text{'}\left(\frac{2}{5}\left(0\right),\frac{2}{5}\left(10\right)\right)\\ \Rightarrow A\text{'}\left(0,10\right)\to A\text{'}\text{'}\left(0,4\right)\end{array}$

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

Again, on dilation with scale factor $k=\frac{2}{5}$

We get

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

  $\begin{array}{l}C\left(10,5\right)\to C\text{'}\left(2\left(10\right),2\left(5\right)\right)\\ \Rightarrow C\left(10,5\right)\to C\text{'}\left(20,10\right)\end{array}$

Again, on dilation with scale factor $k=\frac{2}{5}$

We get

  $\begin{array}{l}C\text{'}\left(20,10\right)\to C\text{'}\text{'}\left(\frac{2}{5}\left(20\right),\frac{2}{5}\left(10\right)\right)\\ \Rightarrow C\text{'}\left(20,10\right)\to C\text{'}\text{'}\left(8,4\right)\end{array}$

Therefore,

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,4\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(8,4\right)\end{array}$

b.

To determine

To find: Vertices of each figure after a dilation.

Answer to Problem 22HP

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,4\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(8,4\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}A=\left(0,5\right)\\ B=\left(0,0\right)\\ C=\left(10,5\right)\end{array}$

  $k=\frac{2}{5}$ followed by $k=2$

First dilation is $\left(x,y\right)\to \left(2x,2y\right)$ .

Multiply the coordinates of each vertex by $\frac{2}{5}$ .

  $\begin{array}{l}A\left(0,5\right)\to A\text{'}\left(\frac{2}{5}\left(0\right),\frac{2}{5}\left(5\right)\right)\\ \Rightarrow A\left(0,5\right)\to A\text{'}\left(0,2\right)\end{array}$

Again, on dilation with scale factor $k=2$

We get

  $\begin{array}{l}A\text{'}\left(0,2\right)\to A\text{'}\text{'}\left(2\left(0\right),2\left(2\right)\right)\\ \Rightarrow A\text{'}\left(0,2\right)\to A\text{'}\text{'}\left(0,4\right)\end{array}$

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

Again, on dilation with scale factor $k=2$

We get

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

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

Again, on dilation with scale factor $k=2$

We get

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

Therefore,

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,4\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(8,4\right)\end{array}$

c.

To determine

To find: Vertices of each figure after a dilation.

Answer to Problem 22HP

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,\frac{5}{2}\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(5,\frac{5}{2}\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}A=\left(0,5\right)\\ B=\left(0,0\right)\\ C=\left(10,5\right)\end{array}$

  $k=\frac{1}{4}$ followed by $k=2$

First dilation is $\left(x,y\right)\to \left(2x,2y\right)$ .

Multiply the coordinates of each vertex by $\frac{1}{4}$ .

  $\begin{array}{l}A\left(0,5\right)\to A\text{'}\left(\frac{1}{4}\left(0\right),\frac{1}{4}\left(5\right)\right)\\ \Rightarrow A\left(0,5\right)\to A\text{'}\left(0,\frac{5}{4}\right)\end{array}$

Again, on dilation with scale factor $k=2$

We get

  $\begin{array}{l}A\text{'}\left(0,\frac{5}{4}\right)\to A\text{'}\text{'}\left(2\left(0\right),2\left(\frac{5}{4}\right)\right)\\ \Rightarrow A\text{'}\left(0,\frac{5}{4}\right)\to A\text{'}\text{'}\left(0,\frac{5}{2}\right)\end{array}$

  $\begin{array}{l}B\left(0,0\right)\to B\text{'}\left(\frac{1}{4}\left(0\right),\frac{1}{4}\left(0\right)\right)\\ \Rightarrow B\left(0,0\right)\to B\text{'}\left(0,0\right)\end{array}$

Again, on dilation with scale factor $k=2$

We get

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

  $\begin{array}{l}C\left(10,5\right)\to C\text{'}\left(\frac{1}{4}\left(10\right),\frac{1}{4}\left(5\right)\right)\\ \Rightarrow C\left(10,5\right)\to C\text{'}\left(\frac{5}{2},\frac{5}{4}\right)\end{array}$

Again, on dilation with scale factor $k=2$

We get

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

Therefore,

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,\frac{5}{2}\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(5,\frac{5}{2}\right)\end{array}$

d.

To determine

To find: Vertices of each figure after a dilation.

Answer to Problem 22HP

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,\frac{5}{2}\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(5,\frac{5}{2}\right)\end{array}$

Explanation of Solution

Given information:

  $\begin{array}{l}A=\left(0,5\right)\\ B=\left(0,0\right)\\ C=\left(10,5\right)\end{array}$

  $k=2$ followed by $k=\frac{1}{4}$

First dilation is $\left(x,y\right)\to \left(2x,2y\right)$ .

Multiply the coordinates of each vertex by $2$ .

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

Again, on dilation with scale factor $k=\frac{1}{4}$

We get

  $\begin{array}{l}A\text{'}\left(0,10\right)\to A\text{'}\text{'}\left(\frac{1}{4}\left(0\right),\frac{1}{4}\left(10\right)\right)\\ \Rightarrow A\text{'}\left(0,10\right)\to A\text{'}\text{'}\left(0,\frac{5}{2}\right)\end{array}$

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

Again, on dilation with scale factor $k=\frac{1}{4}$

We get

  $\begin{array}{l}B\text{'}\left(0,0\right)\to B\text{'}\text{'}\left(\frac{1}{4}\left(0\right),\frac{1}{4}\left(0\right)\right)\\ \Rightarrow B\text{'}\left(0,0\right)\to B\text{'}\text{'}\left(0,0\right)\end{array}$

  $\begin{array}{l}C\left(10,5\right)\to C\text{'}\left(2\left(10\right),2\left(5\right)\right)\\ \Rightarrow C\left(10,5\right)\to C\text{'}\left(20,10\right)\end{array}$

Again, on dilation with scale factor $k=\frac{1}{4}$

We get

  $\begin{array}{l}C\text{'}\left(20,10\right)\to C\text{'}\text{'}\left(\frac{1}{4}\left(20\right),\frac{1}{4}\left(10\right)\right)\\ \Rightarrow C\text{'}\left(20,10\right)\to C\text{'}\text{'}\left(5,\frac{5}{2}\right)\end{array}$

Therefore,

Coordinates after dilation are

  $\begin{array}{l}A\text{'}\text{'}\left(0,\frac{5}{2}\right)\\ B\text{'}\text{'}\left(0,0\right)\\ C\text{'}\text{'}\left(5,\frac{5}{2}\right)\end{array}$