Question

segment C prime D prime has endpoints located at C′(5, 0) and D′(3, 0). It was dilated at a scale factor of 2 from center (3, 0).

Answer 1

C(4, 0)

D(3, 0)

Explanation:

The preimage points can be found by reversing the transformation caused by the dilation. The center of dilation is an invariant point.

Image

For some dilation factor k working about some center O, the point P is transformed to point P' by ...

P' = O + k(P -O)

P' = kP -(k -1)O

Pre-image

Then the reverse transformation can be found by solving for P:

P' +(k -1)O = kP

P = (P' +(k -1)O)/k

For k=2 and O(3, 0), this becomes ...

(x, y) = ((x', y') +(2 -1)(3, 0))/2

(x, y) = (x' +3, y')/2

C = (5 +3, 0)/2 = (4, 0)

D = (3 +3, 0)/2 = (3, 0)

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Additional comment

The center of dilation is invariant. It remains in place regardless of the dilation factor. Here, the center of dilation (3, 0) is the same as the image point D'(3, 0). Hence, the pre-image point D must be in the same location. The math above shows that is the case.

You will notice that C' is twice as far from the center of dilation as is C. This is because of the scale factor of 2.