Triangle ABC had vertices point A(-2,4), point B(-6,8), and point C(-2,4). After dilation, triangle A’B’C’ had vertices point A’(-1,2), point B’(-3,4), and poor C’(-1,2). Determine the scale factor for the dilation of triangle ABC with a center point of dilation at the origin (0,0).
To find the scale factor for the dilation, we can compare the corresponding distances between the corresponding vertices of the two triangles.
The distance between points A(-2, 4) and B(-6, 8) is determined using the distance formula:
d(A, B) = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d(A, B) = sqrt((-6 - (-2))^2 + (8 - 4)^2) = sqrt((-4)^2 + 4^2) = sqrt(16 + 16) = sqrt(32) = 4*sqrt(2)
The distance between points A'(-1, 2) and B'(-3, 4) is:
d(A', B') = sqrt((-3 - (-1))^2 + (4 - 2)^2) = sqrt((-2)^2 + 2^2) = sqrt(4 + 4) = sqrt(8) = 2*sqrt(2)
Therefore, the scale factor for the dilation is:
d(A', B') / d(A, B) = (2*sqrt(2)) / (4*sqrt(2)) = 2 / 4 = 1/2
Hence, the scale factor for the dilation of triangle ABC is 1/2.