Pythagoras Theorem Questions - Free Printable

Let's solve each of these right triangle problems using Pythagoras' Theorem, which states:

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

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

We will apply this to each triangle, identifying which side is missing.

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🔹 Triangle 1

Sides: 1.3 cm, 0.8 cm — find the third side.

Assume it’s a right triangle with legs 1.3 cm and 0.8 cm. Then:

$$
c^2 = 1.3^2 + 0.8^2 = 1.69 + 0.64 = 2.33 \\
c = \sqrt{2.33} \approx 1.5 \text{ cm (to 1dp)}
$$

✔ Answer: 1.5 cm

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🔹 Triangle 2

Sides: 2.5 m, 3.2 m — find the third side.

Here, 3.2 m is longer than 2.5 m, so likely hypotenuse.

Check: Is $ 2.5^2 + x^2 = 3.2^2 $? Or is $ 2.5^2 + 3.2^2 = x^2 $? Let's test both.

Try if 3.2 is hypotenuse:

$$
x^2 = 3.2^2 - 2.5^2 = 10.24 - 6.25 = 3.99 \\
x = \sqrt{3.99} \approx 2.0 \text{ m (to 1dp)}
$$

✔ Answer: 2.0 m

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🔹 Triangle 3

Sides: 1.8 mm, 2 mm — find missing side.

2 mm > 1.8 mm → assume 2 mm is hypotenuse.

$$
x^2 = 2^2 - 1.8^2 = 4 - 3.24 = 0.76 \\
x = \sqrt{0.76} \approx 0.9 \text{ mm (to 1dp)}
$$

✔ Answer: 0.9 mm

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🔹 Triangle 4

Sides: 2.4 cm, 1 cm — find missing side.

2.4 > 1 → likely hypotenuse.

$$
x^2 = 2.4^2 - 1^2 = 5.76 - 1 = 4.76 \\
x = \sqrt{4.76} \approx 2.2 \text{ cm (to 1dp)}
$$

✔ Answer: 2.2 cm

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🔹 Triangle 5

Sides: 2.3 cm, 5 cm — find missing side.

5 cm is clearly the longest → hypotenuse.

$$
x^2 = 5^2 - 2.3^2 = 25 - 5.29 = 19.71 \\
x = \sqrt{19.71} \approx 4.4 \text{ cm (to 1dp)}
$$

✔ Answer: 4.4 cm

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🔹 Triangle 6

Sides: 1.2 m, 2.5 m — find missing side.

2.5 m is longer → likely hypotenuse.

$$
x^2 = 2.5^2 - 1.2^2 = 6.25 - 1.44 = 4.81 \\
x = \sqrt{4.81} \approx 2.2 \text{ m (to 1dp)}
$$

✔ Answer: 2.2 m

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🔹 Triangle 7

Sides: 5.4 cm, 4 cm — find missing side.

5.4 > 4 → likely hypotenuse.

$$
x^2 = 5.4^2 - 4^2 = 29.16 - 16 = 13.16 \\
x = \sqrt{13.16} \approx 3.6 \text{ cm (to 1dp)}
$$

✔ Answer: 3.6 cm

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🔹 Triangle 8

Sides: 1.6 m, 1.2 m — find missing side.

1.6 > 1.2 → likely hypotenuse.

$$
x^2 = 1.6^2 - 1.2^2 = 2.56 - 1.44 = 1.12 \\
x = \sqrt{1.12} \approx 1.0 \text{ m (to 1dp)}
$$

✔ Answer: 1.0 m

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🔹 Triangle 9

Sides: 0.9 mm, 1.1 mm — find missing side.

1.1 > 0.9 → likely hypotenuse.

$$
x^2 = 1.1^2 - 0.9^2 = 1.21 - 0.81 = 0.40 \\
x = \sqrt{0.40} \approx 0.6 \text{ mm (to 1dp)}
$$

✔ Answer: 0.6 mm

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✔ Final Answers (in order):

| Triangle | Missing Side |
|--------|--------------|
| 1 | 1.5 cm |
| 2 | 2.0 m |
| 3 | 0.9 mm |
| 4 | 2.2 cm |
| 5 | 4.4 cm |
| 6 | 2.2 m |
| 7 | 3.6 cm |
| 8 | 1.0 m |
| 9 | 0.6 mm |

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📝 Summary:

We used Pythagoras’ Theorem: $ a^2 + b^2 = c^2 $. For each triangle:
- Identified the longest side as the hypotenuse.
- Used subtraction to find the missing leg: $ x = \sqrt{c^2 - a^2} $
- Rounded all decimal answers to 1 decimal place.

Let me know if you'd like this formatted as a printable answer sheet!