Question

A triangle with vertices at A(20, –30), B(10, –15), and C(5, –20) has been dilated with a center of dilation at the origin. The image of B, point B prime, has the coordinates (2, –3). What is the scale factor of the dilation?
StartFraction 1 Over 10 EndFraction
One-fifth
5
10

Answer 1

To find the scale factor of the dilation, we need to compare the coordinates of point B before and after dilation:

1. Original coordinates of point B: \( (10, -15) \)

2. Dilated coordinates of point B (B'): \( (2, -3) \)

The scale factor \( k \) is determined using the formula:

\[
k = \frac{\text{Image coordinates}}{\text{Original coordinates}}
\]

We can calculate the scale factor using each coordinate:

For the x-coordinate:
\[
k = \frac{2}{10} = \frac{1}{5}
\]

For the y-coordinate:
\[
k = \frac{-3}{-15} = \frac{3}{15} = \frac{1}{5}
\]

Since both coordinates give the same scale factor, the scale factor of the dilation is \( \frac{1}{5} \).

Thus, the answer is **One-fifth**.