How to Find the Missing Side of a Right Triangle

When dealing with a right triangle, finding a missing side is one of the most common mathematical tasks. Whether you are framing a wall or passing a geometry test, knowing how to calculate an unknown leg is crucial.

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Using the Pythagorean Theorem

To find a missing side, you must know the lengths of the other two sides. The rule that connects all three sides is the Pythagorean theorem: a² + b² = c².

In this formula, c is always the longest side (the hypotenuse), while a and b are the shorter legs that meet at the right angle.

Formula to Find Leg "a"

If you know the hypotenuse (c) and Leg b, rearrange the formula to solve for a:

a = √(c² - b²)

Formula to Find Leg "b"

If you know the hypotenuse (c) and Leg a, rearrange the formula to solve for b:

b = √(c² - a²)

Step-by-Step Example

Imagine you have a ladder leaning against a wall. The ladder is 13 feet long (the hypotenuse, c), and the base of the ladder is 5 feet away from the wall (Leg b). How high up the wall does the ladder reach (Leg a)?

1. Identify the knowns: c = 13, b = 5.
2. Apply the formula: a = √(13² - 5²)
3. Square the numbers: a = √(169 - 25)
4. Subtract: a = √(144)
5. Find the square root: a = 12

The ladder reaches exactly 12 feet up the wall.