50 Pythagorean Theorem Worksheet With Answers - Free Printable

The image you've uploaded is a worksheet on Pythagoras' Theorem, which states:

> For any right-angled triangle:
> $$
> a^2 + b^2 = c^2
> $$
> where:
> - $ a $ and $ b $ are the two shorter sides (legs),
> - $ c $ is the hypotenuse (the longest side, opposite the right angle).

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🔷 Part 1: Finding the Hypotenuse

In these problems, we are given the two shorter sides ($ a $ and $ b $), and we need to find the hypotenuse $ c $. We use:

$$
c = \sqrt{a^2 + b^2}
$$

Let’s go through each one:

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#### 1. $ a = 8\text{cm}, b = 6\text{cm} $

$$
c^2 = 8^2 + 6^2 = 64 + 36 = 100 \\
c = \sqrt{100} = 10\text{cm}
$$

✔ Answer: 10 cm

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#### 2. $ a = 12\text{cm}, b = 9\text{cm} $

$$
c^2 = 12^2 + 9^2 = 144 + 81 = 225 \\
c = \sqrt{225} = 15\text{cm}
$$

✔ Answer: 15 cm

---

#### 3. $ a = 8\text{cm}, b = 6\text{cm} $

Same as #1 → $ c = 10\text{cm} $

✔ Answer: 10 cm

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#### 4. $ a = 14\text{cm}, b = 16\text{cm} $

$$
c^2 = 14^2 + 16^2 = 196 + 256 = 452 \\
c = \sqrt{452} \approx 21.26\text{cm}
$$

But the answer shown is 16.6cm, which is incorrect.

Wait — let's check: maybe it's not $ a=14 $, $ b=16 $? But the diagram shows both legs labeled. Wait, perhaps the labeling is off?

Actually, looking at the diagram:
- One leg is 14cm,
- Other leg is 16cm,
- Hypotenuse is $ x $

So yes:
$$
x^2 = 14^2 + 16^2 = 196 + 256 = 452 \\
x = \sqrt{452} \approx 21.26\text{cm}
$$

But the red answer says 16.6cm — that's wrong.

✘ Error in worksheet: The correct answer should be ≈21.3 cm, not 16.6 cm.

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#### 5. $ a = 16\text{cm}, b = 16\text{cm} $

$$
c^2 = 16^2 + 16^2 = 256 + 256 = 512 \\
c = \sqrt{512} \approx 22.63\text{cm}
$$

But the red answer says 18.9cm — again, incorrect.

Wait — maybe this is an isosceles right triangle? Yes, but still:

$$
c = \sqrt{2} \times 16 \approx 1.414 \times 16 \approx 22.62\text{cm}
$$

✘ So 18.9cm is wrong.

---

#### 6. $ a = 17\text{cm}, b = 15\text{cm} $

$$
c^2 = 17^2 + 15^2 = 289 + 225 = 514 \\
c = \sqrt{514} \approx 22.67\text{cm}
$$

But the red answer says 18.8cm — ✘ Incorrect.

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#### 7. $ a = 23\text{cm}, b = 10\text{cm} $

$$
c^2 = 23^2 + 10^2 = 529 + 100 = 629 \\
c = \sqrt{629} \approx 25.08\text{cm}
$$

Red answer says 23.3cm — ✘ Incorrect.

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⚠️ Conclusion: The answers for problems 4–7 under "Finding the Hypotenuse" appear to be incorrect. They do not match the actual calculations.

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🔷 Part 2: Finding a Shorter Side

Here, we know the hypotenuse $ c $ and one leg $ a $ or $ b $, and we need to find the other leg using:

$$
a = \sqrt{c^2 - b^2} \quad \text{(or vice versa)}
$$

Let’s solve each:

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#### 1. $ c = 5\text{cm}, b = 3\text{cm} $, find $ a $

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

✔ Correct! Answer: 4 cm

---

#### 2. $ c = 17\text{cm}, a = 15\text{cm} $, find $ b $

$$
b^2 = c^2 - a^2 = 17^2 - 15^2 = 289 - 225 = 64 \\
b = \sqrt{64} = 8\text{cm}
$$

✔ Correct! Answer: 8 cm

---

#### 3. $ c = 25\text{cm}, a = 24\text{cm} $, find $ b $

$$
b^2 = 25^2 - 24^2 = 625 - 576 = 49 \\
b = \sqrt{49} = 7\text{cm}
$$

✔ Correct! Answer: 7 cm

---

#### 4. $ c = 16\text{cm}, b = 11\text{cm} $, find $ a $

$$
a^2 = 16^2 - 11^2 = 256 - 121 = 135 \\
a = \sqrt{135} \approx 11.62\text{cm}
$$

Answer shown: 11.6cm ✔ Correct (rounded)

---

#### 5. $ c = 19\text{cm}, b = 10\text{cm} $, find $ a $

$$
a^2 = 19^2 - 10^2 = 361 - 100 = 261 \\
a = \sqrt{261} \approx 16.15\text{cm}
$$

Answer shown: 16.2cm ✔ Close enough (rounded)

---

#### 6. $ c = 13\text{cm}, b = 5\text{cm} $, find $ a $

$$
a^2 = 13^2 - 5^2 = 169 - 25 = 144 \\
a = \sqrt{144} = 12\text{cm}
$$

Answer shown: 16.5cm ✘ Incorrect

Wait — the diagram shows $ c = 13 $, $ b = 5 $, and $ a = ? $

But the answer says 16.5cm — that can't be, because $ a < c $ in a right triangle.

So either:
- The diagram is mislabeled,
- Or the answer is wrong.

But $ \sqrt{13^2 - 5^2} = \sqrt{144} = 12 $, so correct answer is 12 cm.

✘ So 16.5cm is wrong.

---

#### 7. $ c = 13.4\text{cm}, b = 12\text{cm} $, find $ a $

$$
a^2 = 13.4^2 - 12^2 = 179.56 - 144 = 35.56 \\
a = \sqrt{35.56} \approx 5.96\text{cm} \approx 6.0\text{cm}
$$

But the red answer says 13.4cm — that's the hypotenuse!

✘ This is clearly wrong.

Wait — maybe they meant $ a = 13.4 $? But no — $ c $ is always the hypotenuse.

So if $ c = 13.4 $, $ b = 12 $, then $ a \approx 6.0 $ cm.

But the answer says 13.4cm — which is already $ c $.

✘ Wrong answer

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✔ Final Summary

| Problem | Type | Given | Correct Answer | Worksheet Answer | Correct? |
|--------|------|-------|----------------|------------------|----------|
| 1 | Hypotenuse | 8, 6 | 10 cm | 10 cm | ✔ |
| 2 | Hypotenuse | 12, 9 | 15 cm | 15 cm | ✔ |
| 3 | Hypotenuse | 8, 6 | 10 cm | 10 cm | ✔ |
| 4 | Hypotenuse | 14, 16 | ~21.3 cm | 16.6 cm | ✘ |
| 5 | Hypotenuse | 16, 16 | ~22.6 cm | 18.9 cm | ✘ |
| 6 | Hypotenuse | 17, 15 | ~22.7 cm | 18.8 cm | ✘ |
| 7 | Hypotenuse | 23, 10 | ~25.1 cm | 23.3 cm | ✘ |
| 1 | Shorter | 5, 3 | 4 cm | 4 cm | ✔ |
| 2 | Shorter | 17, 15 | 8 cm | 8 cm | ✔ |
| 3 | Shorter | 25, 24 | 7 cm | 7 cm | ✔ |
| 4 | Shorter | 16, 11 | ~11.6 cm | 11.6 cm | ✔ |
| 5 | Shorter | 19, 10 | ~16.2 cm | 16.2 cm | ✔ |
| 6 | Shorter | 13, 5 | 12 cm | 16.5 cm | ✘ |
| 7 | Shorter | 13.4, 12 | ~6.0 cm | 13.4 cm | ✘ |

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🚨 Conclusion

- Part 1 (Hypotenuse): Problems 4–7 have incorrect answers.
- Part 2 (Shorter side): Only problems 6 and 7 have wrong answers.

The worksheet contains several errors in the provided solutions.

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✔ Corrected Answers

#### Finding the Hypotenuse
4. $ \sqrt{14^2 + 16^2} = \sqrt{452} \approx 21.3 $ cm
5. $ \sqrt{16^2 + 16^2} = \sqrt{512} \approx 22.6 $ cm
6. $ \sqrt{17^2 + 15^2} = \sqrt{514} \approx 22.7 $ cm
7. $ \sqrt{23^2 + 10^2} = \sqrt{629} \approx 25.1 $ cm

#### Finding a Shorter Side
6. $ \sqrt{13^2 - 5^2} = \sqrt{144} = 12 $ cm
7. $ \sqrt{13.4^2 - 12^2} = \sqrt{35.56} \approx 6.0 $ cm

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Let me know if you'd like a corrected version of the worksheet!