Unknown Side Lengths in Right Triangles Worksheets - Free Printable

To solve the missing side lengths in each right triangle using the Pythagorean theorem, we use the formula:

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

where:
- \( a \) and \( b \) are the legs of the right triangle,
- \( c \) is the hypotenuse (the longest side).

Let's solve each problem step by step.

---

1)

Given:
- One leg = 12 yd
- Hypotenuse = 15 yd
- Missing side = \( x \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
x^2 + 12^2 = 15^2
\]
\[
x^2 + 144 = 225
\]
Subtract 144 from both sides:
\[
x^2 = 81
\]
Take the square root of both sides:
\[
x = \sqrt{81} = 9
\]

Answer:
\[
x = 9 \text{ yd}
\]

---

2)

Given:
- One leg = 16 in
- Hypotenuse = 20 in
- Missing side = \( x \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
x^2 + 16^2 = 20^2
\]
\[
x^2 + 256 = 400
\]
Subtract 256 from both sides:
\[
x^2 = 144
\]
Take the square root of both sides:
\[
x = \sqrt{144} = 12
\]

Answer:
\[
x = 12 \text{ in}
\]

---

3)

Given:
- One leg = 5 ft
- Other leg = 12 ft
- Missing side = \( x \) (hypotenuse)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
5^2 + 12^2 = x^2
\]
\[
25 + 144 = x^2
\]
\[
169 = x^2
\]
Take the square root of both sides:
\[
x = \sqrt{169} = 13
\]

Answer:
\[
x = 13 \text{ ft}
\]

---

4)

Given:
- One leg = 8 yd
- Hypotenuse = 10 yd
- Missing side = \( x \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
x^2 + 8^2 = 10^2
\]
\[
x^2 + 64 = 100
\]
Subtract 64 from both sides:
\[
x^2 = 36
\]
Take the square root of both sides:
\[
x = \sqrt{36} = 6
\]

Answer:
\[
x = 6 \text{ yd}
\]

---

5)

Given:
- One leg = 18 in
- Hypotenuse = 30 in
- Missing side = \( x \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
x^2 + 18^2 = 30^2
\]
\[
x^2 + 324 = 900
\]
Subtract 324 from both sides:
\[
x^2 = 576
\]
Take the square root of both sides:
\[
x = \sqrt{576} = 24
\]

Answer:
\[
x = 24 \text{ in}
\]

---

6)

Given:
- One leg = 36 ft
- Hypotenuse = 65 ft
- Missing side = \( x \)

Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
\[
x^2 + 36^2 = 65^2
\]
\[
x^2 + 1296 = 4225
\]
Subtract 1296 from both sides:
\[
x^2 = 2929
\]
Take the square root of both sides:
\[
x = \sqrt{2929} = 54
\]

Answer:
\[
x = 54 \text{ ft}
\]

---

Final Answers:

\[
\boxed{9, 12, 13, 6, 24, 54}
\]