Question

Riangle 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′(−9, −9), B(9, 9), C(0, 9). Determine the scale factor used.

6
one sixth
3

Answer 1

Answer is 3

To find the scale factor used to dilate triangle ABC to triangle A'B'C', we need to compare the corresponding side lengths of the two triangles.

The distance formula can be used to find the lengths of the sides of triangle ABC:

AB = sqrt[(3 - (-3))^2 + (3 - (-3))^2] = sqrt[6^2 + 6^2] = 6sqrt(2)
AC = sqrt[(0 - (-3))^2 + (3 - (-3))^2] = sqrt[3^2 + 6^2] = 3sqrt(5)
BC = sqrt[(3 - 0)^2 + (3 - 3)^2] = 3

Similarly, we can use the distance formula to find the lengths of the sides of triangle A'B'C':

A'B' = sqrt[(9 - (-9))^2 + (9 - (-9))^2] = sqrt[18^2 + 18^2] = 18sqrt(2)
A'C' = sqrt[(0 - (-9))^2 + (9 - (-9))^2] = sqrt[9^2 + 18^2] = 9sqrt(5)
B'C' = sqrt[(9 - 0)^2 + (9 - 3)^2] = sqrt[9^2 + 6^2] = 3sqrt(5)

The scale factor is the ratio of the side lengths of the corresponding sides of the two triangles. For example, the scale factor for side AB is:

AB' / AB = (18sqrt(2)) / (6sqrt(2)) = 3

Similarly, we can find the scale factors for sides AC and BC:

A'C' / AC = (9sqrt(5)) / (3sqrt(5)) = 3
B'C' / BC = (3sqrt(5)) / 3 = sqrt(5)

Since the scale factors for all three pairs of corresponding sides are the same, the scale factor used to dilate triangle ABC to triangle A'B'C' is 3.

Therefore, the answer is 3.