Working Backwards

Part of the Triangles section of Coddy's Geometry journey. Lesson 35 of 71.

When you know the hypotenuse and one leg, you can use the theorem to find the other leg. Start from a^2 + b^2 = c^2 and subtract b^2 from both sides. The missing leg squared is the hypotenuse squared minus the known leg squared: a^2 = c^2 - b^2. With hypotenuse 10 and leg 8:

$A right triangle with hypotenuse 10, one leg 8 and the other leg a unknown$

Hypotenuse 10, leg 8: how long is a?

a = √(10^2 - 8^2)

a = √(100 - 64)

a = √(36) = 6

To find a leg, subtract the square of the known leg from the square of the hypotenuse. Here 100 - 64 = 36, so the missing leg is 6. Always put the hypotenuse squared first. Subtracting the other way round would give a negative number under the root, and a negative number has no square root. Check the result: 6^2 + 8^2 = 100.