Question

The segment with endpoints (0,8) and (-6,0) is dilated to a segment with endpoints of (0,6) and (-4.5,0). What is the scale factor of the dilation?

Answer 1

To solve the exercise, we can first find the length of each segment using the formula for the distance between two points:

$\begin{gathered} D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{ Where }(x_1,y_1)\text{ and }(x_2,y_2)\text{ are the coordinates of the points} \end{gathered}$

First segment:

$\begin{gathered} (x_1,y_1)=(0,8) \\ (x_2,y_2)=\mleft(-6,0\mright) \\ D=\sqrt[]{(-6_{}-0)^2+(0-8)^2} \\ D=\sqrt[]{(-6)^2+(8)^2} \\ D=\sqrt[]{36+64} \\ D=\sqrt[]{100} \\ D=10 \end{gathered}$

Second segment:

$\begin{gathered} (x_1,y_1)=\mleft(0,6\mright) \\ (x_2,y_2)=\mleft(-4.5,0\mright) \\ D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ D=\sqrt[]{(-4.5-0)^2+(0-6)^2} \\ D=\sqrt[]{(-4.5)^2+(-6)^2} \\ D=\sqrt[]{20.25+36} \\ D=\sqrt[]{56.25} \\ D=7.5 \end{gathered}$

Now, we apply the following formula:

$\frac{\text{image}}{\text{pre}-\text{image}}=\text{scale factor}$

Then, we have:

$\begin{gathered} \frac{\text{ length of second segment}}{\text{ length of first segment}}=\text{scale factor} \\ \frac{7.5}{10}=\text{ scale factor} \\ \text{ Simplify} \\ \frac{7.5\cdot10}{10\cdot10}=\text{ scale factor} \\ \frac{75}{100}=\text{ scale factor} \\ \frac{25\cdot3}{25\cdot4}=\text{ scale factor} \\ \frac{3}{4}=\text{ scale factor} \end{gathered}$

Therefore, the scale factor of the dilation is 3/4.