Question

50 POINTS Triangle ABC with vertices at A(−3, −3), B(3, 3), C(0, 3) is dilated to create triangle A′B′C′ with vertices at A′(−12, −12), B′(12, 12), C′(0, 12). Determine the scale factor used.

9
one nineth
4
one fourth

Answer 1

4,

Explanation:

Answer 2

To determine the scale factor used for the dilation, we can calculate the ratio of the corresponding side lengths of the two triangles.

Let's first find the side lengths of the original triangle ABC:

- AB = sqrt((3-(-3))^2 + (3-(-3))^2) = sqrt(72) = 6sqrt(2)

- BC = sqrt((0-3)^2 + (3-3)^2) = 3

- AC = sqrt((-3-0)^2 + (-3-3)^2) = sqrt(72) = 6sqrt(2)

Now, let's find the side lengths of the dilated triangle A'B'C':

- A'B' = sqrt((12-(-12))^2 + (12-(-12))^2) = sqrt(2(12^2)) = 24sqrt(2)

- B'C' = sqrt((0-12)^2 + (12-3)^2) = sqrt(153)

- A'C' = sqrt((-12-0)^2 + (-12-3)^2) = sqrt(2(153)) = 3sqrt(2) * sqrt(17)

The ratio of corresponding side lengths is:

- A'B' / AB = (24sqrt(2)) / (6sqrt(2)) = 4

- B'C' / BC = sqrt(153) / 3 ≈ 1.732

- A'C' / AC = (3sqrt(2) * sqrt(17)) / (6sqrt(2)) = sqrt(17) / 2 ≈ 2.061

Therefore, the scale factor used for the dilation is 4, since A'B' is 4 times the length of AB.