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
- Apply dilations on the coordinate plane
- Use scale factors from the origin
- Connect dilations to similarity
Worked example
Problem. Dilate point (3, -4) by scale factor 2 centered at the origin.
- Multiply the x-coordinate by 2.
- Multiply the y-coordinate by 2.
- (3, -4) becomes (6, -8).
- Connect the result back to Dilations and Similarity Transformations so the geometric relationship is explicit.
Answer: (6, -8)
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.
Practice problems
1. Dilate (2, 5) by scale factor 3 from the origin.
Choices: (6, 15) · (5, 8) · (3, 10) · (-6, -15)
Show solution
- Multiply both coordinates by 3.
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
- Take half of each coordinate.
Answer: (-2, 3)
3. A side length of 7 is dilated by scale factor 4. What is the new length?
Show solution
- 7 x 4 = 28.
Answer: 28
4. A dilated side length is 18 from an original side length of 6. What is the scale factor?
Show solution
- 18/6 = 3.
Answer: 3
5. A dilation by scale factor 2 creates a figure that is...
Choices: Similar · Congruent only · Perpendicular · Supplementary
Show solution
- Dilations preserve shape but change size.
Answer: Similar
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