Dilations in the Coordinate Plane

First consider dilations with the origin as center.  Then the coordinate rule for a dilation with scale factor k is simply this:

(x, y) --> (kx, ky).

Example 6:  Triangle ABC has coordinates A(–1, –3), B(1, 1) and C(2, –3).  Triangle DEF has coordinates D(2, 6), E(–4, 6) and F(–2, –2).  Show that triangle DFE is the image of triangle ABC under a dilation with center at the origin, and find the scale factor.

Solution:  The image of A is given by (–1, –3) --> (–1k, –3k).  If D is that image, then –1k = 2 and –3k = 6.  Both give k = –2.  If we apply this dilation to B and C, we find that F is the image of B and E is the image of C.

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