MathIsimple

Combined Similarity & Trigonometry

Integrate similarity and trigonometry concepts to solve complex geometric problems involving heights, shadows, and real-world measurements!

10th Grade
Geometry
~60 min
🎮 Interactive Activity: Similarity-Trig Problem Solver

Solve problems combining similarity and trigonometry!

Problem:

Two similar right triangles. In the smaller, opposite = 3, hypotenuse = 5. In the larger, hypotenuse = 10. Find opposite side.
🎮 Interactive Activity: Height Finder

Use trigonometry to find heights!

Problem:

Building casts 30-ft shadow. Angle of elevation to top is 45°. Find building height.
1. Introduction to Combined Concepts

Why Combine Similarity and Trigonometry?

Many real-world problems require both similarity and trigonometry. Understanding how these concepts work together makes problem-solving more powerful and efficient.

Key Connections:

  • Similar triangles have the same trigonometric ratios
  • Trigonometry helps find missing sides in similar triangles
  • Shadow problems combine both concepts naturally
  • Height measurement problems often use both
Example 1: Similar Right Triangles

Two similar right triangles. Smaller: opposite = 3, hypotenuse = 5. Larger: hypotenuse = 10. Find larger opposite side.

Method 1 (Similarity): Scale factor = 10/5 = 2, so opposite = 3 × 2 = 6

Method 2 (Trigonometry): sin(angle) = 3/5 in both. Larger opposite = 10 × (3/5) = 6

Both methods give the same answer!

Example 2: Shadow Problem

A 6-ft person casts 4-ft shadow. Tree casts 20-ft shadow. Find tree height.

Method 1 (Similarity): Scale factor = 20/4 = 5, so height = 6 × 5 = 30 ft

Method 2 (Trigonometry): tan(angle) = 6/4 = 1.5, so height = 20 × 1.5 = 30 ft

2. How to Combine the Concepts
3. Shadow Problems
4. Height Measurement Problems
5. Real-World Applications
6. Problem-Solving Strategies
7. Complex Problems
8. Real-World Scenarios
Frequently Asked Questions

Practice Time!

Practice Quiz
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Questions
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1
Two similar right triangles. Smaller: opposite=3, hypotenuse=5. Larger: hypotenuse=10. Find larger opposite side.
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2
A 6-ft person casts 4-ft shadow. Tree casts 20-ft shadow. Find tree height.
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3
Building casts 30-ft shadow. Angle of elevation is 45°. Find building height.
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4
Two similar triangles have scale factor 2. If sin(angle) = 0.6 in smaller, what is sin(angle) in larger?
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5
Tree height unknown. Shadow 20 ft. Angle of elevation 60°. Find height.
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6
In similar right triangles, if one has sides 3-4-5, and the other has hypotenuse 15, what are the other sides?
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7
A flagpole and its shadow form a right triangle. Shadow is 12 ft, angle of elevation is 30°. Find flagpole height.
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8
Two similar triangles. One has sin(θ) = 0.8. The other has the same angle θ. What is sin(θ) in the second?
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9
A ladder leans against a wall. Base is 5 ft from wall, angle is 60°. Find ladder length.
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10
Combining similarity and trigonometry is useful for:
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