Pythagoras' Triangle Test 1 worksheet featuring nine triangles with side lengths in cm, mm, m, and in, requiring students to apply the Pythagorean theorem to identify right triangles.

Let's solve this step by step using Pythagoras' Theorem.

🔷 Pythagoras’ Theorem:

For a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides:

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

Where:
- $ c $ is the hypotenuse (longest side),
- $ a $ and $ b $ are the other two sides.

We will check each triangle to see if this equation holds (within rounding to 1 decimal place).

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🔹 Triangle 1: Sides 4 cm, 6 cm, 8.5 cm

- Longest side: 8.5 cm → assume hypotenuse
- Check:
$$
4^2 + 6^2 = 16 + 36 = 52 \\
8.5^2 = 72.25
$$
- 52 ≠ 72.25 → ✘ Not a right triangle

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🔹 Triangle 2: Sides 5 mm, 6 mm, 7.8 mm

- Longest side: 7.8 mm
- Check:
$$
5^2 + 6^2 = 25 + 36 = 61 \\
7.8^2 = 60.84 \approx 60.8
$$
- 61 ≈ 60.8 → Close! But not exactly equal, and since decimals are rounded to 1dp, let’s accept small differences.
- But 61 ≠ 60.8, so it's not satisfying the theorem precisely.
- So: ✘ Not a right triangle

> Note: If it were exact, we'd say yes, but here the difference is about 0.2, which is significant for 1dp.

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🔹 Triangle 3: Sides 3 cm, 4 cm, 5 cm

- Classic Pythagorean triple!
- Check:
$$
3^2 + 4^2 = 9 + 16 = 25 \\
5^2 = 25
$$
- ✔ Yes, right triangle

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🔹 Triangle 4: Sides 7 cm, 7 cm, 9.3 cm

- Longest side: 9.3 cm
- Check:
$$
7^2 + 7^2 = 49 + 49 = 98 \\
9.3^2 = 86.49
$$
- 98 ≠ 86.49 → ✘ Not a right triangle

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🔹 Triangle 5: Sides 4 m, 7 m, 8.1 m

- Longest: 8.1 m
- Check:
$$
4^2 + 7^2 = 16 + 49 = 65 \\
8.1^2 = 65.61
$$
- 65 vs 65.61 → very close, but not equal.
- Since decimals are rounded to 1dp, maybe it's acceptable?
- But 65 ≠ 65.61, so even with rounding, it doesn't satisfy.
- ✘ Not a right triangle

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🔹 Triangle 6: Sides 5 cm, 9 cm, 10.3 cm

- Longest: 10.3 cm
- Check:
$$
5^2 + 9^2 = 25 + 81 = 106 \\
10.3^2 = 106.09
$$
- 106 vs 106.09 → extremely close!
- Rounded to 1dp: both are ~106.0 or 106.1
- This could be due to rounding.
- Since $ 10.3^2 = 106.09 $, and $ 5^2 + 9^2 = 106 $, they are almost equal.
- Given that all decimals are rounded to 1dp, we can consider this close enough to be a right triangle.
- ✔ Yes, likely a right triangle (due to rounding)

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🔹 Triangle 7: Sides 9 cm, 12 cm, 15 cm

- Check:
$$
9^2 + 12^2 = 81 + 144 = 225 \\
15^2 = 225
$$
- ✔ Yes, right triangle (another multiple of 3-4-5: 3×3=9, 4×3=12, 5×3=15)

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🔹 Triangle 8: Sides 8 m, 11 m, 12.5 m

- Longest: 12.5 m
- Check:
$$
8^2 + 11^2 = 64 + 121 = 185 \\
12.5^2 = 156.25
$$
- 185 ≠ 156.25 → ✘ Not a right triangle

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🔹 Triangle 9: Sides 6 in, 8 in, 10 in

- Check:
$$
6^2 + 8^2 = 36 + 64 = 100 \\
10^2 = 100
$$
- ✔ Yes, right triangle (another 3-4-5 multiple: 2×3=6, 2×4=8, 2×5=10)

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✔ Final Answers:

| Triangle | Sides | Right Triangle? |
|--------|-------|----------------|
| 1 | 4, 6, 8.5 cm | ✘ No |
| 2 | 5, 6, 7.8 mm | ✘ No |
| 3 | 3, 4, 5 cm | ✔ Yes |
| 4 | 7, 7, 9.3 cm | ✘ No |
| 5 | 4, 7, 8.1 m | ✘ No |
| 6 | 5, 9, 10.3 cm | ✔ Yes (very close, likely due to rounding) |
| 7 | 9, 12, 15 cm | ✔ Yes |
| 8 | 8, 11, 12.5 m | ✘ No |
| 9 | 6, 8, 10 in | ✔ Yes |

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✔ Summary of Right Triangles:

- Triangle 3: 3-4-5
- Triangle 6: 5-9-10.3 (close to perfect)
- Triangle 7: 9-12-15 (3×3-4-5)
- Triangle 9: 6-8-10 (2×3-4-5)

So, 4 triangles are right triangles.

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✔ Final Answer: The right triangles are:
- Triangle 3
- Triangle 6
- Triangle 7
- Triangle 9

All others are not right triangles.