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Similar Triangles - Worksheet 10 - 10th Grade

Hard

Prove triangle similarity using AA, SSS, and SAS (Hard)

23 problems · 25-30 minutes · Skills: triangles, similarity, proportions

About This Worksheet

This hard-level worksheet contains 23 problems designed for 10th Grade students practicing similar triangles. Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Similarity is the gateway to trigonometry. Understanding that similar triangles have proportional sides is the conceptual foundation for defining sine, cosine, and tangent ratios — and it explains why those ratios depend only on the angle, not the size of the triangle.

  • Apply AA, SSS, and SAS similarity criteria to determine if triangles are similar
  • Set up and solve proportions to find missing side lengths in similar triangles
  • Apply the Side-Splitter Theorem and Triangle Proportionality Theorem
  • Write similarity statements listing corresponding vertices in the correct order
  • Use indirect measurement with similar triangles to find heights and distances

How to Solve: Worked Example

Example Problem: Two similar triangles have scale factor 4. The smaller has area 32. Find the larger area.
Solution Steps:
  1. Areas scale by the square of the scale factor, not the factor itself.
  2. The area ratio is 4² = 16.
  3. 32 × 16 = 512.
Answer: 512
Tip: Lengths scale by k, areas by k², volumes by k³. Forgetting to square is the classic error here.
1
Two similar triangles have a scale factor of 2. The smaller has area 15. Find the area of the larger.
2
Two similar triangles have a scale factor of 3. The smaller has area 81. Find the area of the larger.
3
Two similar triangles have a scale factor of 3. The smaller has area 72. Find the area of the larger.
4
Two similar triangles have a scale factor of 2. The smaller has area 28. Find the area of the larger.
5
Two similar triangles have a scale factor of 2. The smaller has area 36. Find the area of the larger.
6
Two similar triangles have a scale factor of 3. The smaller has area 30. Find the area of the larger.
7
Two similar triangles have a scale factor of 2. The smaller has area 15. Find the area of the larger.
8
Two similar triangles have a scale factor of 2. The smaller has area 12. Find the area of the larger.
9
Two similar triangles have a scale factor of 2. The smaller has area 24. Find the area of the larger.
10
Two similar triangles have a scale factor of 2. The smaller has area 14. Find the area of the larger.
11
Two similar triangles have a scale factor of 5. The smaller has area 54. Find the area of the larger.
12
Two similar triangles have a scale factor of 2. The smaller has area 25. Find the area of the larger.
13
Two similar triangles have a scale factor of 3. The smaller has area 30. Find the area of the larger.
14
Two similar triangles have a scale factor of 4. The smaller has area 35. Find the area of the larger.
15
Two similar triangles have a scale factor of 5. The smaller has area 12. Find the area of the larger.
16
Two similar triangles have a scale factor of 3. The smaller has area 27. Find the area of the larger.
17
Two similar triangles have a scale factor of 3. The smaller has area 18. Find the area of the larger.
18
Two similar triangles have a scale factor of 4. The smaller has area 10. Find the area of the larger.
19
Two similar triangles have a scale factor of 2. The smaller has area 8. Find the area of the larger.
20
Two similar triangles have a scale factor of 2. The smaller has area 45. Find the area of the larger.
21
Two similar triangles have a scale factor of 2. The smaller has area 18. Find the area of the larger.
22
Two similar triangles have a scale factor of 2. The smaller has area 15. Find the area of the larger.
23
Two similar triangles have a scale factor of 2. The smaller has area 15. Find the area of the larger.

Answer Key

1. 602. 7293. 6484. 1125. 1446. 2707. 608. 489. 9610. 5611. 135012. 10013. 27014. 56015. 30016. 24317. 16218. 16019. 3220. 18021. 7222. 6023. 60

Frequently Asked Questions

How do I download this similar triangles - worksheet 10 worksheet?

Click the "Download PDF / Print" button at the top of the worksheet. You can save it as a PDF or print it directly. The answer key will be on a separate page.

How do I find worksheets at different difficulty levels?

Each subtopic has 10 worksheets: Worksheets 1-3 are Easy, 4-7 are Medium, and 8-10 are Hard. Start with Easy worksheets to build confidence, then progress to harder ones.

What skills does this triangles worksheet cover?

This worksheet focuses on triangles, similarity, proportions. It is designed for 10th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

Yes. All worksheets on K12Worksheets are free for classroom use. You can print copies for your entire class. With 10 worksheets per subtopic at three difficulty levels, you have plenty of options for differentiated instruction and individualized practice.

How many problems are on this worksheet?

This worksheet contains 23 problems. It typically takes 25-30 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.