Similar Triangles and Scale Factor
A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.
Similar triangles have the same shape but not necessarily the same size. Corresponding angles are congruent, and corresponding side lengths are proportional.
What you'll learn
- Identify similar triangles
- Use scale factors
- Find missing side lengths
Worked example
Problem. Two similar triangles have corresponding sides 4 and 10. A second side in the smaller triangle is 6. Find the matching larger side.
- The scale factor is 10/4 = 2.5.
- Multiply the smaller side by 2.5.
- 6 x 2.5 = 15.
Answer: 15
Practice problems
1. Similar triangles have corresponding angles that are...
Choices: Congruent · Supplementary · Always right · Different
Show solution
- Warm-up: First identify exactly what the question is asking: Similar triangles have corresponding angles that are...
- Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
- Similar figures have equal corresponding angles.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Congruent
2. A side of 5 scales by factor 3. What is the new length?
Show solution
- Warm-up: First identify exactly what the question is asking: A side of 5 scales by factor 3. What is the new length?
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 5 x 3 = 15.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
3. Two similar triangles have sides 4 and 12. What is the scale factor from smaller to larger?
Show solution
- Core Practice: First identify exactly what the question is asking: Two similar triangles have sides 4 and 12. What is the scale factor from smaller to larger?
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 12/4 = 3.
- Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.
Answer: 3
4. A smaller triangle side is 7 and scale factor is 2. What is the larger side?
Show solution
- Core Practice: First identify exactly what the question is asking: A smaller triangle side is 7 and scale factor is 2. What is the larger side?
- Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
- 7 x 2 = 14.
- Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.
Answer: 14
5. Matching sides are 6 and 9. A smaller side is 10. What is the larger matching side?
Show solution
- Challenge: First identify exactly what the question is asking: Matching sides are 6 and 9. A smaller side is 10. What is the larger matching side?
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- Scale factor is 9/6 = 1.5.
- 10 x 1.5 = 15.
- Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.
Answer: 15
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