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} \\ (7.5)/(10)=\text{ scale factor} \\ \text{ Simplify} \\ (7.5\cdot10)/(10\cdot10)=\text{ scale factor} \\ (75)/(100)=\text{ scale factor} \\ (25\cdot3)/(25\cdot4)=\text{ scale factor} \\ (3)/(4)=\text{ scale factor} \end{gathered}`

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