Pythagoras - Right Angle? Determine if each triangle has a right angle based on side lengths.

To determine whether each triangle has a right angle, we use the Pythagorean Theorem:

> For a triangle with sides $ a $, $ b $, and $ c $ (where $ c $ is the longest side),
> it is a right triangle if:
> $$
> a^2 + b^2 = c^2
> $$

We will check each triangle one by one.

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1. Triangle with sides 3, 4, 5

- Check: $ 3^2 + 4^2 = 9 + 16 = 25 $
- $ 5^2 = 25 $
- ✔ $ 25 = 25 $ → Right angle

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2. Triangle with sides 5, 12, 13

- Check: $ 5^2 + 12^2 = 25 + 144 = 169 $
- $ 13^2 = 169 $
- ✔ $ 169 = 169 $ → Right angle

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3. Triangle with sides 6, 7, 13

- Longest side: 13
- Check: $ 6^2 + 7^2 = 36 + 49 = 85 $
- $ 13^2 = 169 $
- ✘ $ 85 \neq 169 $ → Not a right angle

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4. Triangle with sides 3, 7, 7.5

- Check: $ 3^2 + 7^2 = 9 + 49 = 58 $
- $ 7.5^2 = 56.25 $
- ✘ $ 58 \neq 56.25 $ → Not a right angle

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5. Triangle with sides 4, 8, √80

- First, compute $ \sqrt{80} \approx 8.944 $
- So sides are approximately: 4, 8, 8.944 → longest is $ \sqrt{80} $
- Check: $ 4^2 + 8^2 = 16 + 64 = 80 $
- $ (\sqrt{80})^2 = 80 $
- ✔ $ 80 = 80 $ → Right angle

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6. Triangle with sides 8, 24, 26

- Check: $ 8^2 + 24^2 = 64 + 576 = 640 $
- $ 26^2 = 676 $
- ✘ $ 640 \neq 676 $ → Not a right angle

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7. Triangle with sides $ \sqrt{36} $, 8, 10

- $ \sqrt{36} = 6 $
- So sides: 6, 8, 10
- Check: $ 6^2 + 8^2 = 36 + 64 = 100 $
- $ 10^2 = 100 $
- ✔ $ 100 = 100 $ → Right angle

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8. Triangle with sides 11, 12, 17

- Check: $ 11^2 + 12^2 = 121 + 144 = 265 $
- $ 17^2 = 289 $
- ✘ $ 265 \neq 289 $ → Not a right angle

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9. Triangle with sides $ 3^2 $, 14, 15

- $ 3^2 = 9 $
- Sides: 9, 14, 15 → longest is 15
- Check: $ 9^2 + 14^2 = 81 + 196 = 277 $
- $ 15^2 = 225 $
- ✘ $ 277 \neq 225 $ → Not a right angle

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✔ Final Answers:

| Triangle | Has Right Angle? |
|--------|------------------|
| 1. 3, 4, 5 | ✔ Yes |
| 2. 5, 12, 13 | ✔ Yes |
| 3. 6, 7, 13 | ✘ No |
| 4. 3, 7, 7.5 | ✘ No |
| 5. 4, 8, √80 | ✔ Yes |
| 6. 8, 24, 26 | ✘ No |
| 7. √36, 8, 10 | ✔ Yes |
| 8. 11, 12, 17 | ✘ No |
| 9. 9, 14, 15 | ✘ No |

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🟩 Summary:

Triangles with right angles:
→ 1, 2, 5, 7

Triangles without right angles:
→ 3, 4, 6, 8, 9

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