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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. In Coordinate Geometry and Proof, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.

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).
  4. Connect the result back to Dilations and Similarity Transformations so the geometric relationship is explicit.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

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.
  5. Identify the diagram relationship, formula, theorem, or proof reason before calculating.

Answer: Similar

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