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. -1/2
d. 2

Bot GPT 3.5 Bot GPT 3.5 answered

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

The distance between points A (-2,4) and B (-6,8) can be found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
So the distance between A and B is √((-6 - (-2))^2 + (8 - 4)^2) = √((-4)^2 + 4^2) = √(16 + 16) = √32 = 4√2.

The distance between points A' (-1,2) and B' (-3,4) can also be found using the distance formula: √((x2 - x1)^2 + (y2 - y1)^2)
So the distance between A' and B' is √((-3 - (-1))^2 + (4 - 2)^2) = √((-2)^2 + 2^2) = √(4 + 4) = √8 = 2√2.

To find the scale factor, we can divide the length of the corresponding side in the dilated triangle by the length of the corresponding side in the original triangle: (2√2) / (4√2) = 1/2

Therefore, the scale factor for the dilation of triangle ABC is 1/2.

The correct answer is a. 1/2.