6.2: Dilations | Geometry Honors | Code Scholars Tutoring

How it works

A dilation multiplies displacement from its center by a scale factor. Positive factors preserve the direction from the center. Lengths scale by the factor, while areas scale by its square. Factors other than one generally change size.

P\prime=C+k(P-C),\quad k>0

Worked examples

Example 1

Dilate $\displaystyle (3,-2)$ about the origin by factor $\displaystyle 2$ .

The image is $\displaystyle (6,-4)$ .

Example 2

Dilate $\displaystyle (5,3)$ about $\displaystyle C=(1,1)$ by factor $\displaystyle 1/2$ .

The displacement is $\displaystyle (4,2)$ . Half is $\displaystyle (2,1)$ , so the image is $\displaystyle (3,2)$ .

Problems

Problem 6.2.1

Dilate the triangle $\displaystyle (0,0),(3,0),(0,4)$ about the origin by factor $\displaystyle 2$ . Give its image and area.

Triangle with vertices 0,0; 3,0; 0,4. The solution shows its factor-2 image. — Multiply each coordinate by 2.

Show worked solution

The image is $\displaystyle (0,0),(6,0),(0,8)$ . Its area is $\displaystyle 24$ , four times the original area $\displaystyle 6$ .

Triangle with vertices 0,0; 3,0; 0,4. The solution shows its factor-2 image. The solution adds labeled relationships. — The image is (0,0), (6,0), (0,8), with area 24.

Problem 6.2.2

A dilation changes a length from $\displaystyle 8$ to $\displaystyle 12$ . Find the scale factor and area factor.

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$\displaystyle k=12/8=3/2$ . The area factor is $\displaystyle (3/2)^2=9/4$ .

Problem 6.2.3

Can a dilation by factor $\displaystyle 3$ be a rigid motion for a nondegenerate triangle?

Show worked solution

No. Its nonzero side lengths triple, so distances are not preserved.

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 6.2.1

Problem 6.2.2

Problem 6.2.3

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