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
−12
negative Start Fraction 1 over 2 End Fraction
12
Start Fraction 1 over 2 End Fraction
2
2
−2
The scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0) can be determined by finding the ratio of the corresponding side lengths of the original triangle and the dilated triangle.
The distance between points A (-2,4) and B (-6,8) can be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-6 - (-2))^2 + (8 - 4)^2)
= √((-4)^2 + 4^2)
= √(16 + 16)
= √32
Similarly, the distance between points A' (-1,2) and B' (-3,4) can be found:
d' = √((-3 - (-1))^2 + (4 - 2)^2)
= √((-2)^2 + 2^2)
= √(4 + 4)
= √8
The scale factor is the ratio of d' to d:
scale factor = d' / d
= √8 / √32
= √(8/32)
= √(1/4)
= 1/2
Therefore, the scale factor for the dilation of triangle ABC is negative 1/2.