Geometry
Proportional Triangles
Similar triangles have the same shape but different sizes. Their corresponding sides are proportional. Set up a proportion with corresponding sides to find missing lengths.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
Two similar triangles have sides 3, 4, 5 and 6, x, 10. Find x.
Step-by-Step Solution
Set up proportion using corresponding sides: 3/6 = 4/x
Cross-multiply: 3x = 24
x = 8
Answer
x = 8
Problem
Triangle A has sides 5 and 8. Triangle B is similar with shortest side 10. Find the longer side of B.
Step-by-Step Solution
Scale factor: 10/5 = 2
Longer side of B = 8 × 2 = 16
Answer
16
Problem
A 6-ft person casts a 4-ft shadow. A nearby tree casts a 20-ft shadow. How tall is the tree?
Step-by-Step Solution
Set up proportion: 6/4 = h/20
Cross-multiply: 4h = 120
h = 30 feet
Answer
30 feet
Problem
Two similar triangles have perimeters 18 and 30. One side of the smaller is 6. Find the corresponding side of the larger.
Step-by-Step Solution
Scale factor = 30/18 = 5/3.
Corresponding side = 6 × (5/3) = 10.
Answer
10 units
Problem
Triangle A has sides 8, 12, 16. Triangle B is similar with shortest side 2. Find the other sides of B.
Step-by-Step Solution
Scale factor = 2/8 = 1/4.
Middle side: 12 × 1/4 = 3.
Longest side: 16 × 1/4 = 4.
Answer
Sides of B: 2, 3, 4
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Similar triangles: sides 4, 6 and 8, x. Find x.
Scale factor is 3. Original side = 7. New side = ?
Sides 5, 10 and x, 18. Find x.
A 5-ft pole casts a 2-ft shadow. A building casts a 30-ft shadow. How tall?
Similar triangles: 3/9 = 5/x. Find x.
Scale factor is 1/2. Original side = 14. New side = ?
Similar triangles: sides 6, 9 and 10, x. Find x.
A 4-ft child casts a 3-ft shadow. A flagpole casts a 18-ft shadow. How tall?
Triangle A has perimeter 20. Triangle B is similar with scale factor 3. What is B's perimeter?
Similar triangles: 2/8 = 3/x. Find x.
Scale factor is 2/5. Original side = 25. New side = ?
Similar triangles: sides 7, 14 and x, 6. Find x.
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