Pythagorean Theorem Worksheets - Free Printable

This worksheet is about applying the Pythagorean Theorem to find the missing side of right triangles. The Pythagorean Theorem states:

> $$
> a^2 + b^2 = c^2
> $$

Where:
- $ a $ and $ b $ are the legs of the right triangle,
- $ c $ is the hypotenuse (the side opposite the right angle, and the longest side).

Let’s go through each problem step by step and verify the solutions.

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Problem 1:

Triangle with:
- $ a = 3 $
- $ b = 4 $
- $ c = ? $

$$
a^2 + b^2 = c^2 \\
3^2 + 4^2 = c^2 \\
9 + 16 = c^2 \\
25 = c^2 \\
c = \sqrt{25} = 5
$$

✔ Answer: 5

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Problem 2:

Triangle with:
- $ a = 12 $
- $ b = ? $
- $ c = 20 $

$$
a^2 + b^2 = c^2 \\
12^2 + b^2 = 20^2 \\
144 + b^2 = 400 \\
b^2 = 400 - 144 = 256 \\
b = \sqrt{256} = 16
$$

✔ Answer: 16

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Problem 3:

Triangle with:
- $ a = ? $
- $ b = 8 $
- $ c = 10 $

$$
a^2 + b^2 = c^2 \\
a^2 + 8^2 = 10^2 \\
a^2 + 64 = 100 \\
a^2 = 100 - 64 = 36 \\
a = \sqrt{36} = 6
$$

✔ Answer: 6

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Problem 4:

Triangle with:
- $ a = 24 $
- $ b = 18 $
- $ c = ? $

$$
a^2 + b^2 = c^2 \\
24^2 + 18^2 = c^2 \\
576 + 324 = c^2 \\
900 = c^2 \\
c = \sqrt{900} = 30
$$

✔ Answer: 30

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Problem 5:

Triangle with:
- $ a = 16 $
- $ b = ? $
- $ c = 34 $

$$
a^2 + b^2 = c^2 \\
16^2 + b^2 = 34^2 \\
256 + b^2 = 1156 \\
b^2 = 1156 - 256 = 900 \\
b = \sqrt{900} = 30
$$

✔ Answer: 30

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Problem 6:

Triangle with:
- $ a = 18 $
- $ b = 80 $
- $ c = ? $

$$
a^2 + b^2 = c^2 \\
18^2 + 80^2 = c^2 \\
324 + 6400 = c^2 \\
6724 = c^2 \\
c = \sqrt{6724} = 82
$$

✔ Answer: 82

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✔ Final Answers:

1. 5
2. 16
3. 6
4. 30
5. 30
6. 82

All answers on the answer key are correct!

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🔍 Summary:

The Pythagorean Theorem allows us to find any missing side of a right triangle if we know the other two sides. The key steps are:
1. Identify which side is unknown.
2. Plug known values into $ a^2 + b^2 = c^2 $.
3. Solve for the unknown.
4. Take the square root to get the length.

Remember:
- $ c $ is always the hypotenuse (longest side).
- If you're solving for a leg, subtract and take the square root.
- If solving for the hypotenuse, add and take the square root.

Let me know if you'd like this in a printable format or need help with similar problems!