Finding missing sides of right triangles, step by step. Covers finding the hypotenuse, finding a leg, Pythagorean triples, and real-world word problems.
In any right triangle, the relationship between the three sides is:
The hypotenuse is the side directly across from the right angle — it's always the longest side. The other two sides are called legs. The theorem only works on right triangles.
Use this when you know both legs and need to find the hypotenuse \(c\).
Square both legs, add them together, then take the square root: \(c = \sqrt{a^2 + b^2}\)
Use this when you know the hypotenuse and one leg and need to find the other leg.
Rearrange the formula to isolate the missing leg: \(a = \sqrt{c^2 - b^2}\). Subtract the known leg squared from the hypotenuse squared, then take the square root.
These are sets of whole numbers that satisfy \(a^2 + b^2 = c^2\). Recognizing them on sight saves significant time on tests.
The classic. All multiples work too: 6-8-10, 9-12-15
Common on SAT and ACT
Shows up in geometry courses regularly
Using the formula on a non-right triangle. The Pythagorean theorem only works when there's a right angle (90°). If you don't have a right triangle, you need the Law of Cosines instead.
Mixing up which side is \(c\). The hypotenuse is always \(c\) — the side opposite the right angle, always the longest. If you plug a leg in as \(c\), you'll get a wrong answer even if your algebra is perfect.
Adding instead of subtracting when finding a leg. To find a missing leg, rearrange to \(b^2 = c^2 - a^2\). Students sometimes write \(c^2 + a^2\) out of habit. You're subtracting, not adding.
Forgetting to take the square root at the end. \(c^2 = 169\) means \(c = 13\), not 169. The square root is the last step — don't leave it as \(c^2\).
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