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)

Question

The scale factor is

Answer 1

To find the scale factor, we can compare the distances between the corresponding vertices of the two triangles.

The distance between A(2, 1) and B(−2, 2) can be found using the distance formula:
AB = sqrt((2 - (-2))^2 + (1 - 2)^2) = sqrt(16 + 1) = sqrt(17)

The distance between A′(4, 2) and B′(−4, 4) can also be found using the distance formula:
A′B′ = sqrt((4 - (-4))^2 + (2 - 4)^2) = sqrt(64 + 4) = sqrt(68) = 2(sqrt(17))

The scale factor is the ratio of the corresponding distances between the vertices:
Scale factor = A′B′ / AB = 2(sqrt(17)) / sqrt(17) = 2

Therefore, the scale factor for the dilation of triangle ABC is 2.