Practice problems for finding missing side lengths in right triangles using the Pythagorean theorem.

This image presents a worksheet on the Pythagorean Theorem, asking to find the missing length 'x' in six different right triangles. I will solve each problem step-by-step.

The Pythagorean Theorem states that for any right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). The formula is:
`a² + b² = c²`
where `c` is the hypotenuse, and `a` and `b` are the legs.

I will now solve each of the six problems.

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Problem 1:
- Legs: 9 and 12
- Hypotenuse: x
- Using the theorem: `9² + 12² = x²`
- `81 + 144 = x²`
- `225 = x²`
- `x = √225 = 15`

Problem 2:
- Leg: 20
- Hypotenuse: 25
- Other leg: x
- Using the theorem: `x² + 20² = 25²`
- `x² + 400 = 625`
- `x² = 625 - 400 = 225`
- `x = √225 = 15`

Problem 3:
- Leg: 9
- Hypotenuse: 15
- Other leg: x
- Using the theorem: `9² + x² = 15²`
- `81 + x² = 225`
- `x² = 225 - 81 = 144`
- `x = √144 = 12`

Problem 4:
- Leg: 8
- Hypotenuse: 10
- Other leg: x
- Using the theorem: `x² + 8² = 10²`
- `x² + 64 = 100`
- `x² = 100 - 64 = 36`
- `x = √36 = 6`

Problem 5:
- Legs: 3 and 4
- Hypotenuse: x
- Using the theorem: `3² + 4² = x²`
- `9 + 16 = x²`
- `25 = x²`
- `x = √25 = 5`

Problem 6:
- Leg: 15
- Hypotenuse: 25
- Other leg: x
- Using the theorem: `15² + x² = 25²`
- `225 + x² = 625`
- `x² = 625 - 225 = 400`
- `x = √400 = 20`

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Final Answers:

1) x = 15
2) x = 15
3) x = 12
4) x = 6
5) x = 5
6) x = 20