Question

a quadrilateral with vertical at A(4,-4), B(4,-16),C(12,-16), and D(12,-4) has been dilated with a center at the origin.The image of D, point D’ has coordinates (36,-12). What is the scale factor of the dilation?

Answer 1

To find the scale factor of the dilation, we need to compare the distance between the origin and each point in the original quadrilateral to the distance between the origin and the corresponding image point in the dilated quadrilateral.

Let's start with point D. The distance between the origin and point D is:

sqrt((12-0)^2 + (-4-0)^2) = sqrt(144 + 16) = sqrt(160)

The distance between the origin and point D' is:

sqrt((36-0)^2 + (-12-0)^2) = sqrt(1296 + 144) = sqrt(1440)

So the scale factor for the dilation with respect to point D is:

sqrt(1440) / sqrt(160) = 3

Similarly, we can find the scale factor for the other points in the quadrilateral:

For point A, the distance ratio is: sqrt((4-0)^2 + (-4-0)^2) / sqrt((0-0)^2 + (0-0)^2) = sqrt(32) / 0 = undefined (since the origin is the center of dilation and we cannot divide by zero).
For point B, the distance ratio is: sqrt((4-0)^2 + (-16-0)^2) / sqrt((0-0)^2 + (0-0)^2) = 4.
For point C, the distance ratio is: sqrt((12-0)^2 + (-16-0)^2) / sqrt((0-0)^2 + (0-0)^2) = 4.
Since the scale factor is the same for all points, we can conclude that the quadrilateral was dilated by a scale factor of 3 with respect to the origin.