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Dilations and Similarity Transformations

A free Geometry lesson from the “Coordinate Geometry and Proof” unit, with a worked example and practice problems including step-by-step solutions.

A dilation changes size by a scale factor while keeping the same shape. When the center is the origin, multiply each coordinate by the scale factor.

What you'll learn

Why it matters: Architects, map designers, and 3D artists use dilations to resize plans and models without changing shape.

Worked example

Problem. Dilate point (3, -4) by scale factor 2 centered at the origin.

  1. Multiply the x-coordinate by 2.
  2. Multiply the y-coordinate by 2.
  3. (3, -4) becomes (6, -8).

Answer: (6, -8)

Practice problems

1. Dilate (2, 5) by scale factor 3 from the origin.

Choices: (6, 15) · (5, 8) · (3, 10) · (-6, -15)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Dilate (2, 5) by scale factor 3 from the origin.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Multiply both coordinates by 3.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (6, 15)

2. Dilate (-4, 6) by scale factor 1/2 from the origin.

Choices: (-2, 3) · (-8, 12) · (2, -3) · (-4, 3)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Dilate (-4, 6) by scale factor 1/2 from the origin.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Take half of each coordinate.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (-2, 3)

3. A side length of 7 is dilated by scale factor 4. What is the new length?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A side length of 7 is dilated by scale factor 4. What is the new length?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 7 x 4 = 28.
  4. Check the result by substituting or estimating: the response should match 28 and make sense in the original problem.

Answer: 28

4. A dilated side length is 18 from an original side length of 6. What is the scale factor?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A dilated side length is 18 from an original side length of 6. What is the scale factor?
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 18/6 = 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

5. A dilation by scale factor 2 creates a figure that is...

Choices: Similar · Congruent only · Perpendicular · Supplementary

Show solution
  1. Challenge: First identify exactly what the question is asking: A dilation by scale factor 2 creates a figure that is...
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Dilations preserve shape but change size.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Similar

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