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

2
2

−2
negative 2

−12
negative Start Fraction 1 over 2 End Fraction

12

Answer 1

To find the scale factor for the dilation of triangle ABC, we need to compare the lengths of corresponding sides in triangle ABC and triangle A'B'C'.

In triangle ABC, the length of side AB is given by the distance formula:
AB = √[(-6 - (-2))^2 + (8 - 4)^2] = √[(-4)^2 + 4^2] = √[16 + 16] = √32

In triangle A'B'C', the length of side A'B' is given by the distance formula:
A'B' = √[(-3 - (-1))^2 + (4 - 2)^2] = √[(-2)^2 + 2^2] = √[4 + 4] = √8

The scale factor is the ratio of the lengths of the corresponding sides:
scale factor = A'B' / AB = √8 / √32 = √(8/32) = √(1/4) = 1/2

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