Question

What is the scale factor in the dilation if the coordinates of A prime are (–7, 6) and the coordinates of C prime are (–4, 3)?

On a coordinate plane, square A B C D has points (negative 21, 18), (negative 12, 18), (negative 12, 9), (negative 21, 9). Square A prime B prime C prime D prime has points (negative 8, 6), (negative 4, 6), (negative 4, 3), (negative 8, 3).
One-third
One-half
2
3

Answer 1

To find the scale factor of the dilation, we can use the coordinates of point A and point C along with their corresponding points A' and C' after dilation.

Given:
- A = (-21, 18) maps to A' = (-7, 6)
- C = (-12, 9) maps to C' = (-4, 3)

We calculate the scale factor \( k \) by determining the ratio of the distances between the original points and the dilated points.

To find the scale factor for A and A', we calculate:
\[
k = \frac{\text{Distance from the origin for } A'}{\text{Distance from the origin for } A}
= \frac{\sqrt{(-7)^2 + 6^2}}{\sqrt{(-21)^2 + 18^2}}
= \frac{\sqrt{49 + 36}}{\sqrt{441 + 324}}
= \frac{\sqrt{85}}{\sqrt{765}}
= \frac{\sqrt{85}}{\sqrt{85 \cdot 9}}
= \frac{1}{3}
\]

Similarly, for C and C' we have:
\[
k = \frac{\text{Distance from the origin for } C'}{\text{Distance from the origin for } C}
= \frac{\sqrt{(-4)^2 + 3^2}}{\sqrt{(-12)^2 + 9^2}}
= \frac{\sqrt{16 + 9}}{\sqrt{144 + 81}}
= \frac{\sqrt{25}}{\sqrt{225}}
= \frac{5}{15}
= \frac{1}{3}
\]

Since both calculations produce the same scale factor of \( \frac{1}{3} \), we confirm that the scale factor in the dilation is:

**One-third**.