Chapter 11.7, Problem 25HP

To determine

To find:What are the coordinates of that vertex after both dilations.

Answer to Problem 25HP

Coordinates of the Vertex after both dilations is $A\text{'}\text{'}\left(3,6\right)$

Explanation of Solution

Given information:

One vertex of triangle is $A\left(3,6\right)$

First it is dilated with a scale factor of $3$

Then again, Dilated with scale factor of $\frac{1}{3}$

The dilation is $\left(x,y\right)\to \left(kx,ky\right)$

Where $k$ is scale factor.

Here for first dilation $k=3$

We get

  $\begin{array}{l}A\left(3,6\right)\to A\text{'}\left(3\left(3\right),3\left(6\right)\right)\\ \Rightarrow A\left(3,6\right)\to A\text{'}\left(9,18\right)\end{array}$

For next dilation

  $k=\frac{1}{3}$

  $\begin{array}{l}A\text{'}\left(9,18\right)\to A\text{'}\text{'}\left(\frac{1}{3}\left(9\right),\frac{1}{3}\left(18\right)\right)\\ \Rightarrow A\text{'}\left(9,18\right)\to A\text{'}\text{'}\left(3,6\right)\end{array}$

Therefore,

Coordinates of the Vertex after both dilations is $A\text{'}\text{'}\left(3,6\right)$