Pythagorean Theorem Examples with Step-by-Step Solutions

Grade: 8-9 | Topic: Geometry

What You Will Learn

By working through these examples you will become confident at applying the Pythagorean theorem to find missing sides of right triangles, check whether three sides form a right triangle, and solve geometry problems that use right triangles in real-world settings. Each example builds on the last, so work through them in order.

Theory

Quick recap of the formula

The Pythagorean theorem states that in any right triangle:

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

where $a$ and $b$ are the legs (the two shorter sides) and $c$ is the hypotenuse (the longest side, opposite the right angle).

Depending on which side is unknown, you rearrange the formula:

Unknown	Formula
Hypotenuse $c$	$c = \sqrt{a^2 + b^2}$
Leg $a$	$a = \sqrt{c^2 - b^2}$
Leg $b$	$b = \sqrt{c^2 - a^2}$

Strategy for every problem

1. Identify the right angle and label the sides $a$, $b$, and $c$.
2. Decide whether you are finding the hypotenuse (add) or a leg (subtract).
3. Substitute the known values into the formula.
4. Simplify and take the square root.
5. Check your answer: the hypotenuse must always be the longest side.

Worked Examples

Example 1: Finding the hypotenuse -- whole number answer (easy)

Problem: A right triangle has legs $a = 5$ cm and $b = 12$ cm. Find the hypotenuse.

Step 1: Write the formula for the hypotenuse. $c = \sqrt{a^2 + b^2}$

Step 2: Substitute the known values. $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}$

Step 3: Take the square root. $c = 13 \text{ cm}$

Answer: The hypotenuse is 13 cm. (This is the well-known 5-12-13 Pythagorean triple.)

Example 2: Finding the hypotenuse -- irrational answer (easy)

Problem: A right triangle has legs $a = 4$ m and $b = 7$ m. Find the hypotenuse.

Step 1: Apply the formula. $c = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}$

Step 2: Since 65 is not a perfect square, leave the answer in radical form or approximate. $c = \sqrt{65} \approx 8.06 \text{ m}$

Answer: The hypotenuse is $\sqrt{65} \approx$ 8.06 m.

Example 3: Finding a missing leg (medium)

Problem: A right triangle has a hypotenuse of $c = 20$ cm and one leg of $b = 16$ cm. Find the other leg.

Step 1: Write the formula solved for the missing leg. $a = \sqrt{c^2 - b^2}$

Step 2: Substitute. $a = \sqrt{20^2 - 16^2} = \sqrt{400 - 256} = \sqrt{144}$

Step 3: Simplify. $a = 12 \text{ cm}$

Answer: The missing leg is 12 cm.

Quick check: Is the hypotenuse still the longest side? $20 > 16$ and $20 > 12$. Yes.

Example 4: Is this a right triangle? (medium)

Problem: A triangle has sides of length 9, 40, and 41. Determine whether it is a right triangle.

Step 1: Identify the longest side as the potential hypotenuse: $c = 41$, $a = 9$, $b = 40$.

Step 2: Check the Pythagorean equation. $a^2 + b^2 = 9^2 + 40^2 = 81 + 1600 = 1681$ $c^2 = 41^2 = 1681$

Step 3: Compare. $1681 = 1681 \checkmark$

Answer: Yes, 9-40-41 is a right triangle because $a^2 + b^2 = c^2$.

Example 5: Diagonal of a rectangle (challenging)

Problem: A rectangular garden measures 15 m by 20 m. A gardener wants to run a path diagonally from one corner to the opposite corner. How long is the diagonal path?

Step 1: A rectangle's diagonal divides it into two right triangles. The length and width become the legs, and the diagonal is the hypotenuse. $a = 15 \text{ m}, \quad b = 20 \text{ m}$

Step 2: Apply the Pythagorean theorem. $d = \sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625}$

Step 3: Simplify. $d = 25 \text{ m}$

Answer: The diagonal path is 25 m long. (Notice that 15-20-25 is the 3-4-5 triple scaled by 5.)

Common Mistakes

Mistake 1: Confusing which side is the hypotenuse

❌ Given a right triangle with sides 8, 15, and 17, using 15 as the hypotenuse: $8^2 + 17^2 = 64 + 289 = 353 \neq 15^2 = 225$. The equation fails.

✅ The hypotenuse is always the longest side: $c = 17$. Check: $8^2 + 15^2 = 64 + 225 = 289 = 17^2$. Correct.

Why this matters: Mislabeling the hypotenuse leads to incorrect subtraction and wrong answers. Always pick the longest side for $c$.

Mistake 2: Forgetting the square root at the end

❌ $c^2 = 25 + 144 = 169$, so $c = 169$.

✅ $c = \sqrt{169} = 13$.

Why this matters: The formula gives $c^2$, not $c$. Stopping before the square root gives an answer that is the square of the actual length.

Mistake 3: Subtracting when finding the hypotenuse

❌ $c = \sqrt{12^2 - 5^2} = \sqrt{144 - 25} = \sqrt{119}$.

✅ $c = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$.

Why this matters: You add when finding the hypotenuse and subtract when finding a leg. Subtracting gives a smaller value than the legs, which is impossible for a hypotenuse.

Practice Problems

Try these on your own before checking the answers:

1. A right triangle has legs of 7 cm and 24 cm. Find the hypotenuse.
2. A right triangle has a hypotenuse of 26 m and one leg of 10 m. Find the other leg.
3. Do the sides 6, 8, and 11 form a right triangle?
4. A right triangle has legs of 3 and 5. Find the hypotenuse in simplified radical form.
5. A football field is 100 yards long and 53.3 yards wide. What is the distance diagonally from one corner to the opposite corner? (Round to one decimal place.)

Click to see answers

1. $c = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$ cm.
2. $a = \sqrt{26^2 - 10^2} = \sqrt{676 - 100} = \sqrt{576} = 24$ m.
3. Longest side is 11. Check: $6^2 + 8^2 = 36 + 64 = 100$. But $11^2 = 121$. Since $100 \neq 121$, this is not a right triangle. (It is obtuse because $a^2 + b^2 < c^2$.)
4. $c = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}$. (34 has no perfect-square factors, so $\sqrt{34}$ is already simplified.)
5. $d = \sqrt{100^2 + 53.3^2} = \sqrt{10{,}000 + 2{,}840.89} = \sqrt{12{,}840.89} \approx 113.3$ yards.

Summary

• Always identify the hypotenuse first -- it is the longest side and opposite the right angle.
• Add squares when finding the hypotenuse: $c = \sqrt{a^2 + b^2}$.
• Subtract squares when finding a leg: $a = \sqrt{c^2 - b^2}$.
• To verify a right triangle, check whether $a^2 + b^2 = c^2$ for the three given sides.
• Answers are not always whole numbers. Leave irrational results in radical form or round as the problem requires.

Related Topics

• Pythagorean Theorem -- Formula, Proof, and Examples
• Pythagorean Theorem Word Problems with Answers
• Distance Formula -- How to Find Distance Between Two Points

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