Pythagorean Theorem practice worksheet with examples and exercises for calculating missing sides of right-angled triangles.

To solve the problems using the Pythagorean theorem, we use the formula:

\[
a^2 + b^2 = c^2
\]

where \( a \) and \( b \) are the lengths of the two shorter sides (legs), and \( c \) is the length of the hypotenuse. Depending on what is missing, we rearrange the formula to solve for the unknown side.

Problem 1: Find \( c \)

Given:
- \( a = 8 \) cm
- \( b = 15 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 8^2 + 15^2
\]
\[
c^2 = 64 + 225
\]
\[
c^2 = 289
\]
\[
c = \sqrt{289}
\]
\[
c = 17 \text{ cm}
\]

Problem 2: Find \( c \)

Given:
- \( a = 16 \) cm
- \( b = 30 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 16^2 + 30^2
\]
\[
c^2 = 256 + 900
\]
\[
c^2 = 1156
\]
\[
c = \sqrt{1156}
\]
\[
c = 34 \text{ cm}
\]

Problem 3: Find \( c \)

Given:
- \( a = 40 \) cm
- \( b = 75 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
c^2 = 40^2 + 75^2
\]
\[
c^2 = 1600 + 5625
\]
\[
c^2 = 7225
\]
\[
c = \sqrt{7225}
\]
\[
c = 85 \text{ cm}
\]

Problem 4: Find \( b \)

Given:
- \( a = 15 \) cm
- \( c = 25 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
25^2 = 15^2 + b^2
\]
\[
625 = 225 + b^2
\]
\[
b^2 = 625 - 225
\]
\[
b^2 = 400
\]
\[
b = \sqrt{400}
\]
\[
b = 20 \text{ cm}
\]

Problem 5: Find \( b \)

Given:
- \( a = 18 \) cm
- \( c = 30 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
30^2 = 18^2 + b^2
\]
\[
900 = 324 + b^2
\]
\[
b^2 = 900 - 324
\]
\[
b^2 = 576
\]
\[
b = \sqrt{576}
\]
\[
b = 24 \text{ cm}
\]

Problem 6: Find \( b \)

Given:
- \( a = 20 \) cm
- \( c = 52 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
52^2 = 20^2 + b^2
\]
\[
2704 = 400 + b^2
\]
\[
b^2 = 2704 - 400
\]
\[
b^2 = 2304
\]
\[
b = \sqrt{2304}
\]
\[
b = 48 \text{ cm}
\]

Problem 7: Find \( a \)

Given:
- \( b = 48 \) cm
- \( c = 60 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
60^2 = a^2 + 48^2
\]
\[
3600 = a^2 + 2304
\]
\[
a^2 = 3600 - 2304
\]
\[
a^2 = 1296
\]
\[
a = \sqrt{1296}
\]
\[
a = 36 \text{ cm}
\]

Problem 8: Find \( a \)

Given:
- \( b = 28 \) cm
- \( c = 35 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
35^2 = a^2 + 28^2
\]
\[
1225 = a^2 + 784
\]
\[
a^2 = 1225 - 784
\]
\[
a^2 = 441
\]
\[
a = \sqrt{441}
\]
\[
a = 21 \text{ cm}
\]

Problem 9: Find \( a \)

Given:
- \( b = 60 \) cm
- \( c = 68 \) cm

Using the Pythagorean theorem:
\[
c^2 = a^2 + b^2
\]
\[
68^2 = a^2 + 60^2
\]
\[
4624 = a^2 + 3600
\]
\[
a^2 = 4624 - 3600
\]
\[
a^2 = 1024
\]
\[
a = \sqrt{1024}
\]
\[
a = 32 \text{ cm}
\]

Final Answers:

\[
\boxed{
\begin{array}{ccc}
1. & c = 17 \text{ cm} & \\
2. & c = 34 \text{ cm} & \\
3. & c = 85 \text{ cm} & \\
4. & b = 20 \text{ cm} & \\
5. & b = 24 \text{ cm} & \\
6. & b = 48 \text{ cm} & \\
7. & a = 36 \text{ cm} & \\
8. & a = 21 \text{ cm} & \\
9. & a = 32 \text{ cm} & \\
\end{array}
}
\]