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 )
.(1 point)
The scale factor is

Answer 1

To determine the scale factor for the dilation of triangle ABC to triangle A'B'C', we can use the coordinates of the points before and after the dilation.

The center of dilation is at the origin (0,0), so the scale factor \( k \) can be calculated for each vertex by using the following formula:

\[
k = \frac{\text{dilated coordinate}}{\text{original coordinate}}
\]

We can analyze each vertex:

1. For point A:
- Original point A: \( (2, 1) \)
- Dilated point A': \( (4, 2) \)
\[
k = \frac{4}{2} = 2 \quad \text{and} \quad k = \frac{2}{1} = 2
\]

2. For point B:
- Original point B: \( (-2, 2) \)
- Dilated point B': \( (-4, 4) \)
\[
k = \frac{-4}{-2} = 2 \quad \text{and} \quad k = \frac{4}{2} = 2
\]

3. For point C:
- Original point C: \( (-3, -1) \)
- Dilated point C': \( (-6, -2) \)
\[
k = \frac{-6}{-3} = 2 \quad \text{and} \quad k = \frac{-2}{-1} = 2
\]

In all cases, the scale factor is the same. Thus, the scale factor for the dilation is:

\[
\boxed{2}
\]