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) . (1 point)

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) . (1 point)
Responses

A) −1/2

B) −2

C) 2

D) 1/2

GPT 3.5 · Answer

To find the scale factor of the dilation, we can compare the lengths of corresponding sides of the two triangles.

The length of side AB in triangle ABC is given by the distance formula: √((-6 - (-2))^2 + (8 - 4)^2) = √(4^2 + 4^2) = √32 = 4√2.

The length of side A'B' in triangle A'B'C' is given by the distance formula: √((-3 - (-1))^2 + (4 - 2)^2) = √(2^2 + 2^2) = √8 = 2√2.

Therefore, the scale factor of the dilation is 2√2 / 4√2 = 1/2.

So, the correct answer is D) 1/2.