Special Right Triangles Worksheet: Calculate unknown sides in given right triangles.

Let's solve each of these special right triangles step by step. There are two types of special right triangles:

1. 45°-45°-90° triangle:
- The legs are equal: $ a = a $
- Hypotenuse = $ a\sqrt{2} $

2. 30°-60°-90° triangle:
- Short leg (opposite 30°) = $ a $
- Long leg (opposite 60°) = $ a\sqrt{3} $
- Hypotenuse = $ 2a $

We'll apply these rules to each problem.

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Problem 1

- Triangle: 45°-45°-90°
- Given: hypotenuse = $ 2\sqrt{8} $
- Simplify: $ 2\sqrt{8} = 2 \cdot 2\sqrt{2} = 4\sqrt{2} $
- In a 45-45-90 triangle, hypotenuse = leg × $ \sqrt{2} $
- So: $ x = y $, and $ x\sqrt{2} = 4\sqrt{2} $
- Divide both sides: $ x = 4 $
- So: $ x = 4 $, $ y = 4 $

✔ Answer: $ x = 4 $, $ y = 4 $

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Problem 2

- Triangle: 45°-45°-90°
- Given: one leg = $ 2\sqrt{4} $
- Simplify: $ \sqrt{4} = 2 $, so $ 2\sqrt{4} = 2 \cdot 2 = 4 $
- So one leg is 4 → other leg $ y = 4 $
- Hypotenuse $ x = 4\sqrt{2} $

✔ Answer: $ x = 4\sqrt{2} $, $ y = 4 $

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Problem 3

- Triangle: 45°-45°-90°
- Given: hypotenuse = 6
- So: $ x\sqrt{2} = 6 $ → $ x = \frac{6}{\sqrt{2}} = \frac{6\sqrt{2}}{2} = 3\sqrt{2} $
- Since it's 45-45-90, both legs are equal: $ x = y = 3\sqrt{2} $

✔ Answer: $ x = 3\sqrt{2} $, $ y = 3\sqrt{2} $

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Problem 4

- Triangle: 30°-60°-90°
- Given: short leg (opposite 30°) = $ \sqrt{3} $
- So: short leg = $ a = \sqrt{3} $
- Long leg (opposite 60°) = $ a\sqrt{3} = \sqrt{3} \cdot \sqrt{3} = 3 $
- Hypotenuse = $ 2a = 2\sqrt{3} $
- So: $ x = $ long leg = 3, $ y = $ hypotenuse = $ 2\sqrt{3} $

✔ Answer: $ x = 3 $, $ y = 2\sqrt{3} $

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Problem 5

- Triangle: 30°-60°-90°
- Given: side opposite 30° = $ 10\sqrt{6} $? Wait — let’s check angles:
- Right angle at bottom-left
- 60° at top, 30° at bottom-right
- So the side adjacent to 30° is the longer leg, which is $ 10\sqrt{6} $
- In 30-60-90:
- Side opposite 30° = $ a $
- Side opposite 60° = $ a\sqrt{3} $
- Hypotenuse = $ 2a $
- Here, the side opposite 60° is $ 10\sqrt{6} $, so:
$$
a\sqrt{3} = 10\sqrt{6}
$$
Solve for $ a $:
$$
a = \frac{10\sqrt{6}}{\sqrt{3}} = 10 \cdot \sqrt{2} = 10\sqrt{2}
$$
So:
- $ x = a = 10\sqrt{2} $ (side opposite 30°)
- $ y = 2a = 20\sqrt{2} $ (hypotenuse)

✔ Answer: $ x = 10\sqrt{2} $, $ y = 20\sqrt{2} $

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Problem 6

- Triangle: 45°-45°-90°
- Given: hypotenuse = 28
- So: $ x\sqrt{2} = 28 $ → $ x = \frac{28}{\sqrt{2}} = \frac{28\sqrt{2}}{2} = 14\sqrt{2} $
- Both legs are equal: $ x = y = 14\sqrt{2} $

✔ Answer: $ x = 14\sqrt{2} $, $ y = 14\sqrt{2} $

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Problem 7

- Triangle: 45°-45°-90°
- Given: hypotenuse = $ 10\sqrt{2} $
- So: $ x\sqrt{2} = 10\sqrt{2} $ → $ x = 10 $
- Since it's 45-45-90, both legs are equal: $ x = y = 10 $

✔ Answer: $ x = 10 $, $ y = 10 $

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Problem 8

- Triangle: 30°-60°-90°
- Given: side opposite 30° = 12
- So: $ a = 12 $
- Then:
- Long leg (opposite 60°) = $ a\sqrt{3} = 12\sqrt{3} $
- Hypotenuse = $ 2a = 24 $
- So: $ x = $ long leg = $ 12\sqrt{3} $, $ y = $ hypotenuse = 24

✔ Answer: $ x = 12\sqrt{3} $, $ y = 24 $

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✔ Final Answers Summary:

| Problem | $ x $ | $ y $ |
|--------|-------------------|---------------------|
| 1 | 4 | 4 |
| 2 | $ 4\sqrt{2} $ | 4 |
| 3 | $ 3\sqrt{2} $ | $ 3\sqrt{2} $ |
| 4 | 3 | $ 2\sqrt{3} $ |
| 5 | $ 10\sqrt{2} $ | $ 20\sqrt{2} $ |
| 6 | $ 14\sqrt{2} $ | $ 14\sqrt{2} $ |
| 7 | 10 | 10 |
| 8 | $ 12\sqrt{3} $ | 24 |

Let me know if you'd like this in printable format or with diagrams!