Chapter 11.8, Problem 18HP

a.

To determine

To find:If two triangles are similar, determine what two transformations map $\Delta DEF$ onto $\Delta D\text{'}E\text{'}F\text{'}$

Answer to Problem 18HP

First $\Delta DEF$ is rotated about ${90}^0$

And the dilated with a scale factor of $k=3$ to get $\Delta D\text{'}E\text{'}F\text{'}$

Explanation of Solution

Given information:

Two triangles $\Delta DEF$ and $\Delta D\text{'}E\text{'}F\text{'}$ are similar

Coordinates of $\Delta D\text{'}E\text{'}F\text{'}$ are

  $\begin{array}{l}D\text{'}=\left(3,3\right)\\ E\text{'}=\left(6,3\right)\\ F\text{'}=\left(3,-6\right)\end{array}$

Graph of Both triangles are

  

From graph

We can say that

First $\Delta DEF$ is rotated about ${90}^0$

And the dilated with a scale factor

As given both triangles are similar

Scale factor will ratio of any side of Image divided by corresponding side of Original triangle.

From graph we can say that

  $\begin{array}{l}DE=1\\ D\text{'}E\text{'}=3\end{array}$

Therefore,

Scale factor $=\frac{D\text{'}E\text{'}}{DE}=\frac{3}{1}=3$

Therefore,

First $\Delta DEF$ is rotated about ${90}^0$

And the dilated with a scale factor of $k=3$ to get $\Delta D\text{'}E\text{'}F\text{'}$

b.

To determine

To find: The Coordinates of the vertices of the image.

Answer to Problem 18HP

Vertices of the image are

  $\begin{array}{l}D\text{'}\text{'}\left(3,13\right)\\ E\text{'}\text{'}\left(6,13\right)\\ F\text{'}\text{'}\left(3,4\right)\end{array}$

Explanation of Solution

Given information:

Coordinates of $\Delta D\text{'}E\text{'}F\text{'}$ are

  $\begin{array}{l}D\text{'}=\left(3,3\right)\\ E\text{'}=\left(6,3\right)\\ F\text{'}=\left(3,-6\right)\end{array}$

  $\Delta D\text{'}E\text{'}F\text{'}$ translated Up ten units.

Given

Coordinates of $\Delta D\text{'}E\text{'}F\text{'}$ are

  $\begin{array}{l}D\text{'}=\left(3,3\right)\\ E\text{'}=\left(6,3\right)\\ F\text{'}=\left(3,-6\right)\end{array}$

  $\Delta D\text{'}E\text{'}F\text{'}$ translated Up ten units.

Translated Up ten units means y-coordinate of each vertex should be added with $10$ .

Therefore,

New vertices are

  $\begin{array}{l}D\text{'}\left(3,3\right)\to D\text{'}\text{'}\left(3,3+10\right)\\ \Rightarrow D\text{'}\left(3,3\right)\to D\text{'}\text{'}\left(3,13\right)\\ E\text{'}\left(6,3\right)\to E\text{'}\text{'}\left(6,3+10\right)\\ \Rightarrow E\text{'}\left(6,3\right)\to E\text{'}\text{'}\left(6,13\right)\\ F\text{'}\left(3,-6\right)\to F\text{'}\text{'}\left(3,-6+10\right)\\ \Rightarrow F\text{'}\left(3,-6\right)\to F\text{'}\text{'}\left(3,4\right)\end{array}$

Therefore,

Vertices of the image are

  $\begin{array}{l}D\text{'}\text{'}\left(3,13\right)\\ E\text{'}\text{'}\left(6,13\right)\\ F\text{'}\text{'}\left(3,4\right)\end{array}$