Pythagoras' Theorem practice worksheet with nine right triangles, each with two side lengths provided, requiring students to calculate the missing side length using the Pythagorean theorem.

Let’s solve each triangle one by one using the Pythagorean Theorem.

The Pythagorean Theorem says:
In a right triangle, if the two shorter sides are a and b, and the longest side (hypotenuse) is c, then:
> a² + b² = c²

We’ll use this to find the missing side in each triangle. Remember — the hypotenuse is always the longest side, opposite the right angle.

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Triangle 1 (Top Left):

Sides given: 0.8 cm and 1.3 cm → these are the two legs (since 1.3 > 0.8, but we don’t know yet which is hypotenuse). Wait — actually, look at the shape: the right angle is between the 0.8 cm and the unknown side? No — let’s think differently.

Actually, in all these diagrams, the right angle is implied by the shape — the two shorter sides form the right angle, and the longest side is the hypotenuse.

But wait — in some triangles, the labeled sides might include the hypotenuse. We need to check which side is missing.

Looking at Triangle 1: It has sides 0.8 cm and 1.3 cm shown, and one side missing. Since 1.3 cm is longer than 0.8 cm, it could be the hypotenuse or a leg. But in the diagram, the 1.3 cm side looks like the hypotenuse (longest side), and 0.8 cm is one leg. So the missing side is the other leg.

So:
Let a = 0.8 cm
Let c = 1.3 cm (hypotenuse)
Find b.

Formula:
a² + b² = c²
→ 0.8² + b² = 1.3²
→ 0.64 + b² = 1.69
→ b² = 1.69 - 0.64 = 1.05
→ b = √1.05 ≈ 1.0247 → round to 1 decimal place → 1.0 cm

Wait — let me double-check that calculation:

0.8 squared = 0.64
1.3 squared = 1.69
1.69 - 0.64 = 1.05
√1.05 = ? Let's calculate:
1.024695... → yes, rounds to 1.0 cm

✔ Correct.

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Triangle 2 (Top Middle):

Sides: 2.5 m and 3.2 m — again, likely the two legs? Or is 3.2 the hypotenuse?

Looking at the diagram — the 3.2 m side is drawn as the longest side, so probably hypotenuse. Missing side is the other leg.

So:
a = 2.5 m
c = 3.2 m
Find b.

a² + b² = c²
2.5² + b² = 3.2²
6.25 + b² = 10.24
b² = 10.24 - 6.25 = 3.99
b = √3.99 ≈ 1.9975 → rounds to 2.0 m

Check:
√3.99 — since 2²=4, so √3.99 is just under 2 → 1.9975 → yes, 2.0 when rounded to 1 dp.

✔ Correct.

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Triangle 3 (Top Right):

Sides: 1.8 mm and 2 mm — 2 mm is longer, so likely hypotenuse.

Missing side is the other leg.

a = 1.8 mm
c = 2 mm
Find b.

a² + b² = c²
1.8² + b² = 2²
3.24 + b² = 4
b² = 4 - 3.24 = 0.76
b = √0.76 ≈ 0.8718 → rounds to 0.9 mm

Check:
√0.76 — 0.87^2 = 0.7569, close to 0.76 → yes, 0.8718 → rounds to 0.9

✔ Correct.

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Triangle 4 (Middle Left):

Sides: 2.4 cm and 1 cm — 2.4 is longer, so likely hypotenuse.

Missing side is the other leg.

a = 1 cm
c = 2.4 cm
Find b.

a² + b² = c²
1² + b² = 2.4²
1 + b² = 5.76
b² = 5.76 - 1 = 4.76
b = √4.76 ≈ 2.1817 → rounds to 2.2 cm

Check:
2.18^2 = 4.7524, 2.19^2=4.7961 → 4.76 is closer to 2.18 → but 2.1817 rounds to 2.2? Wait — 2.1817 to 1 decimal place: look at second decimal is 8, which is ≥5, so round up first decimal: 2.2 → yes.

✔ Correct.

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Triangle 5 (Middle Center):

Sides: 5 cm and 2.3 cm — 5 cm is much longer, so likely hypotenuse.

Missing side is the other leg.

a = 2.3 cm
c = 5 cm
Find b.

a² + b² = c²
2.3² + b² = 5²
5.29 + b² = 25
b² = 25 - 5.29 = 19.71
b = √19.71 ≈ 4.4396 → rounds to 4.4 cm

Check:
4.4^2 = 19.36, 4.5^2=20.25 → 19.71 is closer to 4.44 → 4.4396 → rounds to 4.4

✔ Correct.

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Triangle 6 (Middle Right):

Sides: 2.5 m and 1.2 m — 2.5 is longer, so likely hypotenuse.

Missing side is the other leg.

a = 1.2 m
c = 2.5 m
Find b.

a² + b² = c²
1.2² + b² = 2.5²
1.44 + b² = 6.25
b² = 6.25 - 1.44 = 4.81
b = √4.81 ≈ 2.193 → rounds to 2.2 m

Check:
2.19^2 = 4.7961, 2.20^2=4.84 → 4.81 is closer to 2.19 → but 2.193 rounds to 2.2? Second decimal is 9, so yes — 2.2

✔ Correct.

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Triangle 7 (Bottom Left):

Sides: 5.4 cm and 4 cm — 5.4 is longer, so likely hypotenuse.

Missing side is the other leg.

a = 4 cm
c = 5.4 cm
Find b.

a² + b² = c²
4² + b² = 5.4²
16 + b² = 29.16
b² = 29.16 - 16 = 13.16
b = √13.16 ≈ 3.6277 → rounds to 3.6 cm

Check:
3.6^2 = 12.96, 3.7^2=13.69 → 13.16 is closer to 3.63 → 3.6277 → rounds to 3.6

✔ Correct.

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Triangle 8 (Bottom Middle):

Sides: 1.6 m and 1.2 m — 1.6 is longer, so likely hypotenuse.

Missing side is the other leg.

a = 1.2 m
c = 1.6 m
Find b.

a² + b² = c²
1.2² + b² = 1.6²
1.44 + b² = 2.56
b² = 2.56 - 1.44 = 1.12
b = √1.12 ≈ 1.0583 → rounds to 1.1 m

Check:
1.05^2 = 1.1025, 1.06^2=1.1236 → 1.12 is very close to 1.058 → rounds to 1.1

✔ Correct.

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Triangle 9 (Bottom Right):

Sides: 0.9 mm and 1.1 mm — 1.1 is longer, so likely hypotenuse.

Missing side is the other leg.

a = 0.9 mm
c = 1.1 mm
Find b.

a² + b² = c²
0.9² + b² = 1.1²
0.81 + b² = 1.21
b² = 1.21 - 0.81 = 0.40
b = √0.40 ≈ 0.6325 → rounds to 0.6 mm

Check:
0.63^2 = 0.3969, 0.64^2=0.4096 → 0.40 is closer to 0.63 → but 0.6325 rounds to 0.6? First decimal is 6, second is 3 < 5 → so should be 0.6? Wait — 0.6325 to 1 decimal place: look at the hundredths digit — 3, which is less than 5, so we do not round up → 0.6

But wait — 0.6325 is closer to 0.6 than to 0.7? Actually, no — 0.6325 is greater than 0.6, and to one decimal place, we look at the next digit: 3 → so we keep 0.6.

Yes — 0.6 mm

✔ Correct.

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Now, let’s list all answers clearly:

1. Top Left: 1.0 cm
2. Top Middle: 2.0 m
3. Top Right: 0.9 mm
4. Middle Left: 2.2 cm
5. Middle Center: 4.4 cm
6. Middle Right: 2.2 m
7. Bottom Left: 3.6 cm
8. Bottom Middle: 1.1 m
9. Bottom Right: 0.6 mm

Final Answer:
1.0 cm, 2.0 m, 0.9 mm, 2.2 cm, 4.4 cm, 2.2 m, 3.6 cm, 1.1 m, 0.6 mm