Apply the Pythagorean theorem to find missing side lengths. | Right triangles and trigonometry | SAT Math

Use a^2 + b^2 = c^2 to find a missing side in any right triangle.

Core Idea

In a right triangle, $𝑎 2 + 𝑏 2 = 𝑐 2$ , where $𝑐$ is the hypotenuse (the side opposite the right angle). This works in both directions: find the hypotenuse from two legs, or find a leg from the hypotenuse and the other leg.

Understanding

Rule: In a right triangle, the two legs satisfy $𝑎 2 + 𝑏 2 = 𝑐 2$ .

$𝑐$ is always the hypotenuse.
Use the Pythagorean triple if one fits.
If you solve for a side, take the positive square root.

The hypotenuse is the side opposite the right angle.

Step by Step

Identify the right angle and label the hypotenuse $𝑐$ (the side across from it).
Label the two legs $𝑎$ and $𝑏$ .
Plug the two known values into $𝑎 2 + 𝑏 2 = 𝑐 2$ .
Solve for the unknown and simplify (take the positive square root for a length).

Misconceptions

Putting a leg value into $𝑐$ instead of the hypotenuse — this makes the equation wrong.
Forgetting to take the square root at the end (giving $𝑐 2$ instead of $𝑐$ ).
Adding the squares when you should subtract (when solving for a leg: $𝑎 2 = 𝑐 2 - 𝑏 2$ ).

Question

Worked Example

In a right triangle, the two legs measure 7 and 24. What is the length of the hypotenuse?

Select an answer to see the explanation