Question

Triangle ABC with vertices at A(−4, −4), B(8, 8), C(2, 6) is dilated to create triangle A′B′C′ with vertices at A′(−2, −2), B′(4, 4), C′(1, 3). Determine the scale factor used.

2
one half
4
one fourth

Answer 1

To determine the scale factor used in the dilation of triangle ABC to triangle A′B′C′, we can compare the coordinates of corresponding vertices.

We have the initial vertices of triangle ABC and the dilated vertices of triangle A′B′C′:

- A(-4, -4) to A′(-2, -2)
- B(8, 8) to B′(4, 4)
- C(2, 6) to C′(1, 3)

To find the scale factor \( k \), we can calculate it by comparing the coordinates:

1. For vertex A and A′:
\[
k = \frac{x'}{x} = \frac{-2}{-4} = \frac{1}{2}
\]

2. For vertex B and B′:
\[
k = \frac{x'}{x} = \frac{4}{8} = \frac{1}{2}
\]

3. For vertex C and C′:
\[
k = \frac{x'}{x} = \frac{1}{2} \quad \text{(for x-coordinates)}
\]
and for the y-coordinates:
\[
k = \frac{3}{6} = \frac{1}{2}
\]

In all cases, we find that the scale factor \( k \) is \( \frac{1}{2} \).

Thus, the scale factor used is **one half**.