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A dilation is a nonrigid transformation that can produce enlarged or reduced images from a given pre-image. How do you know that a dilation will produce similar figures? Explain.
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A dilation with center O and scale factor k sends each point P to P' on ray OP so that OP' = k·OP. This multiplies every position vector relative to O by k, so the image of any segment AB satisfies
A' = O + k(A − O), B' = O + k(B − O) => A'B' = k(A − B).
Taking lengths gives |A'B'| = |k|·|AB|, so every corresponding side length is multiplied by the same constant |k|. For angles, use dot products:
A'B' · A'C' = k^2 (AB · AC),
and dividing by the lengths |A'B'||A'C'| = |k||AB|·|k||AC| cancels the k’s, so the cosine of each angle is unchanged.
Thus a dilation multiplies all side lengths by the same factor and preserves all angles; therefore the image and preimage are similar (except the degenerate k = 0 case, which sends everything to a point).
A' = O + k(A − O), B' = O + k(B − O) => A'B' = k(A − B).
Taking lengths gives |A'B'| = |k|·|AB|, so every corresponding side length is multiplied by the same constant |k|. For angles, use dot products:
A'B' · A'C' = k^2 (AB · AC),
and dividing by the lengths |A'B'||A'C'| = |k||AB|·|k||AC| cancels the k’s, so the cosine of each angle is unchanged.
Thus a dilation multiplies all side lengths by the same factor and preserves all angles; therefore the image and preimage are similar (except the degenerate k = 0 case, which sends everything to a point).
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