Dilation Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

Point

P(8, 12)

is dilated from the origin with scale factor

k = \dfrac{1}{4}

. Find the image

P'

and compare the distance from the origin to

P'

vs. to

P

1
Step 1: Apply $(x, y) \to (kx, ky)$ with $k = \tfrac{1}{4}$ : $P' = \left(\tfrac{1}{4}\cdot 8,\; \tfrac{1}{4}\cdot 12\right) = (2, 3)$ .
2
Step 2: Distance $OP = \sqrt{8^2+12^2} = \sqrt{64+144} = \sqrt{208} = 4\sqrt{13}$ .
3
Step 3: Distance $OP' = \sqrt{2^2+3^2} = \sqrt{13}$ .
4
Step 4: Ratio: $OP'/OP = \sqrt{13}/(4\sqrt{13}) = 1/4 = k$ . The distance from the origin scales by $k$ .

Answer

P'(2, 3)

;

OP' = \sqrt{13}

, which is

\tfrac{1}{4}

OP = 4\sqrt{13}

A scale factor

0 < k < 1

produces a reduction. The image is closer to the origin by factor

k

, so all distances from the origin shrink by the same factor. This confirms dilation is a similarity transformation preserving shape but not size.

About Dilation

A transformation that enlarges or shrinks a figure by a scale factor from a center point.

Triangle [formula] has vertices [formula], [formula], [formula]. Apply a dilation from the origin wi

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