The Pythagorean theorem and special right triangles
The Pythagorean theorem and its converse, classifying triangles as acute, right or obtuse, Pythagorean triples, and 45-45-90 and 30-60-90 triangles.
The theorem
Geometrically: build a square on each side of the right triangle. The area of the square on the hypotenuse equals the combined area of the squares on the legs.
To find the hypotenuse, add the squares; to find a leg, subtract:
Whole-number solutions like \((3, 4, 5)\),\((5, 12, 13)\),\((8, 15, 17)\) and \((7, 24, 25)\) are called Pythagorean triples. Any multiple of a triple, such as \((6, 8, 10)\), is also a triple.
The converse, and acute vs. obtuse
Special right triangles
Two right triangles show up so often that it's worth memorizing their side ratios.
Practice
Quick check
A right triangle has legs \(8\) and \(15\). How long is the hypotenuse?
\(\sqrt{64 + 225} = \sqrt{289} = 17\).
Quick check
The hypotenuse of a 45-45-90 triangle is \(10\). How long is each leg?
\(x\sqrt{2} = 10\), so \(x = \frac{10}{\sqrt{2}} = 5\sqrt{2}\).
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