Pythagorean Triples.

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Presentation transcript:

Pythagorean Triples

Fact In a right triangle, the sides touching the right angle are called legs. The side opposite the right angle is the hypotenuse.

The Pythagorean Theorem, a2 + b2 = c2, relates the sides of RIGHT triangles. a and b are the lengths of the legs and c is the length of the hypotenuse.

A Pythagorean Triple… Is a set of three whole numbers that satisfy the Pythagorean Theorem. What numbers can you think of that would be a Pythagorean Triple? Remember, it has to satisfy the equation a2 + b2 = c2.

Pythagorean Triple: The set {3, 4, 5} is a Pythagorean Triple. a2 + b2 = c2 32 + 42 = 52 9 + 16 = 25 25 = 25

Show that {5, 12, 13} is a Pythagorean triple. Always use the largest value as c in the Pythagorean Theorem. a2 + b2 = c2 52 + 122 = 132 25 + 144 = 169 169 = 169

Show that {2, 2, 5} is not a Pythagorean triple. a2 + b2 = c2 22 + 22 = 52 4 + 4 = 25 8 ≠ 25 Showing that three numbers are a Pythagorean triple proves that the triangle with these side lengths will be a right triangle.

How to find more Pythagorean Triples If we multiply each element of the Pythagorean triple, such as {3, 4, 5} by another integer, like 2, the result is another Pythagorean triple {6, 8, 10}. a2 + b2 = c2 62 + 82 = 102 36 + 64 = 100 100 = 100

By knowing Pythagorean triples, you can quickly solve for a missing side of certain right triangles. Find the length of side b in the right triangle below. Use the Pythagorean triple {5, 12, 13}. The length of side b is 5 units.

Find the length of side a in the right triangle below. It is not obvious which Pythagorean triple these sides represent. Begin by dividing the given sides by their GCF (greatest common factor) { ___, 16, 20} ÷ 4 { ___, 4, 5} We see this is a {3, 4, 5} Pythagorean triple. Since we divided by 4, we must now do the opposite and multiple by 4. {3, 4, 5} • 4 = {12, 16, 20} Side a is 12 units long.