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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
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.