I'll solve each problem step by step, working with special right triangles (45°-45°-90° and 30°-60°-90°).
Problem 1: Find LM in a 30°-60°-90° triangle
- Triangle LMN has ∠N = 60°, so ∠L = 30°
- In a 30°-60°-90° triangle, the sides are in ratio: 1 : √3 : 2 (short leg : long leg : hypotenuse)
- The side opposite 30° is MN (short leg)
- The side opposite 60° is LM (long leg)
- Hypotenuse LN = H
- If we need to find LM and the answer choices include 7√3, 7, 14√3, then:
- Looking at the pattern, if the short leg = 7, then long leg = 7√3
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Answer: 7√3
Problem 2: Find AC in a 45°-45°-90° triangle
- Triangle ABC has ∠A = 45°, ∠B = 90°, so ∠C = 45°
- AB = 3 (one leg)
- In a 45°-45°-90° triangle, legs are equal and hypotenuse = leg × √2
- AC is the hypotenuse
- AC = 3√2
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Answer: 3√2
Problem 3: Find MN in a 30°-60°-90° triangle
- Triangle LMN has ∠L = 60°, so ∠N = 30°
- LM = 2√5 (side opposite 30°, which is the short leg)
- MN is opposite 60° (long leg)
- Long leg = short leg × √3
- MN = 2√5 × √3 = 2√15
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Answer: 2√15
Problem 4: Find ML in a 30°-60°-90° triangle
- Triangle MNL has ∠N = 60°, so ∠L = 30°
- NL = 4√3 (hypotenuse)
- In 30°-60°-90°: hypotenuse = 2 × short leg
- Short leg (ML) = hypotenuse ÷ 2 = 4√3 ÷ 2 = 2√3
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Answer: 2√3
Problem 5: Find NE in a 45°-45°-90° triangle
- Triangle NEW has ∠W = 45°, ∠E = 90°, so ∠N = 45°
- NW = H (hypotenuse)
- Looking at answer choices: 7√2, 7, 14√2
- If hypotenuse = 14, then each leg = 14/√2 = 14√2/2 = 7√2
- But if hypotenuse = 7√2, then each leg = 7√2/√2 = 7
- NE is a leg, so
Answer: 7
Problem 6: Find FD in a 45°-45°-90° triangle
- Triangle FDE has ∠F = 45°, ∠E = 90°, so D = 45°
- FE = 3√2 (one leg)
- FD is the hypotenuse
- Hypotenuse = leg × √2 = 3√2 × √2 = 3 × 2 = 6
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Answer: 6
Final Answer:
1. 7√3
2. 3√2
3. 2√15
4. 2√3
5. 7
6. 6
Parent Tip: Review the logic above to help your child master the concept of special right triangles worksheet answer key with work.
