Asked by Pinapple

Triangle ABC
has vertices point A(2, 1)
, point B(−2, 2)
, and point C(−3,−1)
. After dilation, triangle A′B′C′
has vertices point A′(4, 2)
, point B′(−4, 4)
, and point C′(−6,−2)
. Determine the scale factor for the dilation of triangle ABC
with a center point of dilation at the origin (0, 0 )

Answers

Answered by GPT-4o mini
To determine the scale factor of the dilation of triangle ABC to triangle A'B'C', we need to compare the coordinates of corresponding vertices before and after dilation.

1. **Calculate the distances from the origin to the points of triangle ABC:**
- For point A(2, 1):
\[
\text{Distance from A to origin} = \sqrt{2^2 + 1^2} = \sqrt{4 + 1} = \sqrt{5}
\]
- For point B(−2, 2):
\[
\text{Distance from B to origin} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2}
\]
- For point C(−3, −1):
\[
\text{Distance from C to origin} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10}
\]

2. **Calculate the distances from the origin to the points of triangle A'B'C':**
- For point A'(4, 2):
\[
\text{Distance from A' to origin} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}
\]
- For point B'(-4, 4):
\[
\text{Distance from B' to origin} = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}
\]
- For point C'(-6, -2):
\[
\text{Distance from C' to origin} = \sqrt{(-6)^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10}
\]

3. **Determine the scale factor \( k \):**
The scale factor for a dilation centered at the origin can be found by comparing the distance from the origin to a point before and after the dilation:
\[
k = \frac{\text{Distance from the origin to the dilated point}}{\text{Distance from the origin to the original point}}
\]

We can check the scale factor using any of the corresponding points. Let's use point A and A':

\[
k = \frac{\text{Distance from A' to origin}}{\text{Distance from A to origin}} = \frac{2\sqrt{5}}{\sqrt{5}} = 2
\]

By repeating the process for the other pairs of points (B with B' and C with C'), you will find the same scale factor of \( k = 2 \).

**Final answer:** The scale factor for the dilation is \( \boxed{2} \).
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