Pythagorean Theorem: Formula and Examples

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

• $a$ and $b$ are the legs, the two sides that form the right angle.
• $c$ is the hypotenuse, the longest side, directly across from the right angle.

The theorem only works for right triangles. The converse also works: if $a^2 + b^2 = c^2$ for three side lengths, the triangle has a right angle. For example, sides 9, 12, 15 form a right triangle because $81 + 144 = 225 = 15^2$. Sides 6, 7, 9 do not, because $36 + 49 = 85$, not 81.

When you know both legs, add their squares, then take the square root.

Example 1: The legs are 6 and 8. Find $c$.

1. Square the legs: $6^2 = 36$ and $8^2 = 64$.
2. Add: $36 + 64 = 100$.
3. Take the square root: $c = \sqrt{100} = 10$.

Example 2: The legs are 5 and 7. Find $c$.

1. $5^2 + 7^2 = 25 + 49 = 74$.
2. $c = \sqrt{74} \approx 8.60$.

Not every answer is a whole number. Leave it as $\sqrt{74}$ or round as your teacher asks.

When you know the hypotenuse and one leg, subtract instead of add:

$$b^2 = c^2 - a^2$$

Example 1: The hypotenuse is 13 and one leg is 5. Find the other leg.

1. Square both: $13^2 = 169$ and $5^2 = 25$.
2. Subtract: $169 - 25 = 144$.
3. Square root: $b = \sqrt{144} = 12$.

Example 2: The hypotenuse is 10 and one leg is 4.

1. $b^2 = 100 - 16 = 84$.
2. $b = \sqrt{84} = 2\sqrt{21} \approx 9.17$.

Quick sanity check: a leg must always be shorter than the hypotenuse.

A Pythagorean triple is a set of three whole numbers that fit $a^2 + b^2 = c^2$. Memorizing a few saves time on tests.

Triple	Check
3, 4, 5	$9 + 16 = 25$
5, 12, 13	$25 + 144 = 169$
8, 15, 17	$64 + 225 = 289$
7, 24, 25	$49 + 576 = 625$

Any multiple of a triple is also a triple. Since 3, 4, 5 works, so do 6, 8, 10 and 9, 12, 15. If you see legs of 15 and 20, you can spot that it is 3, 4, 5 times 5, so the hypotenuse is 25.

The distance formula is just the Pythagorean theorem on a coordinate grid:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The horizontal change is one leg, the vertical change is the other leg, and the distance is the hypotenuse.

Example: Find the distance from $(1, 2)$ to $(7, 10)$.

1. Horizontal change: $7 - 1 = 6$.
2. Vertical change: $10 - 2 = 8$.
3. $d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

Problem: A 17-foot ladder leans against a wall. The bottom of the ladder is 8 feet from the wall. How high up the wall does the ladder reach?

1. Draw it. The wall and ground form the right angle. The ladder is the hypotenuse, so $c = 17$. The ground distance is one leg, $a = 8$.
2. Set up: $8^2 + b^2 = 17^2$.
3. Simplify: $64 + b^2 = 289$.
4. Solve: $b^2 = 225$, so $b = 15$.

Answer: The ladder reaches 15 feet up the wall. (This is the 8, 15, 17 triple.) Paste it into the VeilGPT solver to check your steps on similar problems.