Question

Triangle ABC has vertices point A (−2,4) , point B (−6,8) , and point C (−2,4) . After dilation, triangle A′B′C′ has vertices point A′ (−1,2) , point B′ (−3,4) , and point C′ (−1,2) . Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) .

Answer 1

To find the scale factor for the dilation of triangle ABC to triangle A′B′C′ with a center of dilation at the origin (0,0), we can use the relationship between the original coordinates and the coordinates after dilation.

For triangle ABC:
- Point A (−2, 4)
- Point B (−6, 8)
- Point C (−2, 4)

For triangle A′B′C′:
- Point A′ (−1, 2)
- Point B′ (−3, 4)
- Point C′ (−1, 2)

We calculate the scale factor \( k \) for each vertex by using the formula:

\[
k = \frac{\text{New Coordinate}}{\text{Old Coordinate}}
\]

Let’s calculate the scale factor for point A and point A′:

For point A:
- A = (−2, 4)
- A′ = (−1, 2)

Calculating the x-coordinate scale factor:
\[
k_x = \frac{-1}{-2} = \frac{1}{2}
\]

Calculating the y-coordinate scale factor:
\[
k_y = \frac{2}{4} = \frac{1}{2}
\]

Both x and y coordinates give the same scale factor of \( \frac{1}{2} \).

Now let's check point B and point B′ to verify that the same scale factor applies:

For point B:
- B = (−6, 8)
- B′ = (−3, 4)

Calculating the x-coordinate scale factor:
\[
k_x = \frac{-3}{-6} = \frac{1}{2}
\]

Calculating the y-coordinate scale factor:
\[
k_y = \frac{4}{8} = \frac{1}{2}
\]

So, for point B the scale factor is also \( \frac{1}{2} \).

Since the dilations for points A, B, and C yield the same scale factor, we can conclude that the overall scale factor for the dilation of triangle ABC to triangle A′B′C′ is:

\[
\boxed{\frac{1}{2}}
\]