How to solve right triangle problems using Pythagorean triples

On SAT you might see math questions which ask about one side of right triangle. Though you can apply Pythagorean Theorem for solving right triangle, you can also use Pythagorean Triples which allow you to determine a side of the right triangle without involving calculations.

Pythagorean Triples are values that satisfy equation (c2 = a2+b2). For example, (3,4,5)

52 = 32+42
25 = 25

It is important to mention that multiple of this Triple (3,4,5) will also satisfy the equation. For example, (6,8,10), (9,12,15) (12,16,20) will also satisfy the Pythagorean Theorem.

Pythagorean Triples are important because they help you to solve right triangles without calculation.

Example

A right triangle has a leg of 12 and a hypotenuse of 15.What is the length of the other leg?

Solution. You can also calculate second leg by using Pythagorean equation (c2 = a2+b2) but don’t do that. Observe that (x,12,15) is obtained by multiplying (3,4,5) by 3.

So               x = 3 x 3 = 9

Memorizing  a few of the smallest Pythagorean Triples will enable you to solve Right Triangles without calculation:

 (3,4,5) 32 + 42 = 52 (6,8,10) 62 + 82 = 102 (5,12,13) 52 + 122 = 132 (7,24,25) 72 +242 = 252 (8,15,17) 82 +152 = 172

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