Answer key: Pythagorean Theorem Worksheet
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Goal: Use the Pythagorean theorem to find the hypotenuse or a missing leg of a right triangle, and apply it to a real-world problem.
Completed results table
| Setting | Leg b | Hypotenuse c | Area | Angle A |
|---|---|---|---|---|
| Legs 3 and 4 | 4 | 5 | 6 units² | 36.87° |
| Legs 6 and 8 | 8 | 10 | 24 units² | 36.87° |
| Legs 5 and 12 | 12 | 13 | 30 units² | 22.62° |
| Legs 1 and 1 | 1 | 1.414 | 0.5 units² | 45° |
| Leg 9, hypotenuse 15 | 12 | 15 | 54 units² | 36.87° |
| Leg 5, hypotenuse 13 | 12 | 13 | 30 units² | 22.62° |
Answers
Check the 5–12 triangle by hand with a² + b² = c².
5² + 12² = 25 + 144 = 169 = 13², so the hypotenuse is 13.
Compare the 3–4–5 and 6–8–10 triangles. What stays the same, and what changes?
The angles stay the same (angle A ≈ 36.87° in both), because 6–8–10 is the 3–4–5 triangle scaled by 2. Every side doubles, and the area becomes 4 times as large, from 6 to 24 square units.
The legs 1 and 1 give a hypotenuse of about 1.414. What is its exact value?
c² = 1² + 1² = 2, so c = √2 ≈ 1.414. √2 is irrational: its decimal never ends and never repeats.
A 13-foot ladder leans against a wall with its foot 5 feet from the wall. How high up the wall does it reach?
b = √(13² − 5²) = √(169 − 25) = √144 = 12 feet. The last row of the table is this ladder.
Find the missing leg when a = 9 and c = 15. Which famous triangle is this a multiple of?
b = √(225 − 81) = √144 = 12. The 9–12–15 triangle is the 3–4–5 triangle scaled by 3.
Can a leg ever be longer than the hypotenuse? Try a = 20 and c = 15 in missing-leg mode, and explain the message.
No. Because c² = a² + b² and b is positive, c² is bigger than a², so c is bigger than a: the hypotenuse is always the longest side. The calculator says the hypotenuse must be longer than the known leg.