Question

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)

Answer 1

To determine the scale factor for the dilation of triangle \( ABC \) to triangle \( A'B'C' \) with the center of dilation at the origin (0,0), we need to compare the coordinates of the corresponding points before and after dilation.

1. **Calculate the coordinates before and after dilation:**
- \( A(2, 1) \) dilates to \( A'(4, 2) \)
- \( B(-2, 2) \) dilates to \( B'(-4, 4) \)
- \( C(-3, -1) \) dilates to \( C'(-6, -2) \)

2. **Find the scale factor:**
The formula for dilation centered at the origin can be represented as:
\[
(x, y) \rightarrow (kx, ky)
\]
where \( k \) is the scale factor.

We can calculate \( k \) for each corresponding point:

- For point \( A \):
\[
k = \frac{x'}{x} = \frac{4}{2} = 2
\]
\[
k = \frac{y'}{y} = \frac{2}{1} = 2
\]

- For point \( B \):
\[
k = \frac{x'}{x} = \frac{-4}{-2} = 2
\]
\[
k = \frac{y'}{y} = \frac{4}{2} = 2
\]

- For point \( C \):
\[
k = \frac{x'}{x} = \frac{-6}{-3} = 2
\]
\[
k = \frac{y'}{y} = \frac{-2}{-1} = 2
\]

3. **Conclusion:**
In all cases, the scale factor \( k \) is consistent and equal to \( 2 \).

Therefore, the scale factor for the dilation of triangle \( ABC \) to triangle \( A'B'C' \) is \( \boxed{2} \).