7. Triples and Special Right Triangles

Some right triangles have whole-number side lengths called Pythagorean triples. Knowing the common ones saves time:

• \(3, 4, 5\)
• \(5, 12, 13\)
• \(8, 15, 17\)
• \(7, 24, 25\)

Multiples also work: \(6, 8, 10\) is just \(2 \times (3,4,5)\), and \(9, 12, 15\) is \(3 \times (3,4,5)\).

Two special patterns also appear in geometry:

• A 45-45-90 right triangle has two equal legs.
• A 30-60-90 right triangle has a short leg that is half the hypotenuse.

Use these as shortcuts only when the pattern is clearly present. If you are unsure, the Pythagorean Theorem always works.

Common mistake: forcing a triple when the numbers do not fit. Legs \(2\) and \(3\) give \(\sqrt{13}\), not a whole number.