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)
1/2
-2
-1/2
2
To find the scale factor of the dilation, we need to compare the side lengths of triangle ABC and triangle A'B'C'.
The side lengths of triangle ABC are:
AB = sqrt((-6 - (-2))^2 + (8 - 4)^2) = sqrt(16 + 16) = sqrt(32)
BC = sqrt((-2 - (-6))^2 + (4 - 8)^2) = sqrt(16 + 16) = sqrt(32)
CA = sqrt((-2 - (-2))^2 + (4 - 4)^2) = sqrt(0 + 0) = 0
The side lengths of triangle A'B'C' are:
A'B' = sqrt((-3 - (-1))^2 + (4 - 2)^2) = sqrt(4 + 4) = sqrt(8)
B'C' = sqrt((-1 - (-3))^2 + (2 - 4)^2) = sqrt(4 + 4) = sqrt(8)
C'A' = sqrt((-2 - (-1))^2 + (4 - 2)^2) = sqrt(1 + 4) = sqrt(5)
To find the scale factor, we can compare the ratios of the corresponding side lengths:
AB' / AB = sqrt(8) / sqrt(32) = sqrt(8/32) = sqrt(1/4) = 1/2
BC' / BC = sqrt(8) / sqrt(32) = sqrt(8/32) = sqrt(1/4) = 1/2
CA' / CA = sqrt(5) / 0 (since we can't divide by zero) = undefined
Since AB' / AB = 1/2 and BC' / BC = 1/2, we can conclude that the scale factor for the dilation is 1/2.
Therefore, the correct answer is 1/2.