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Pythagorean Triples

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 ,. a 2 + b 2 = c 2 , relates the sides of RIGHT triangles.

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Pythagorean Triples

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  1. Pythagorean Triples

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

  3. 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.

  4. 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.

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

  6. 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

  7. 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.

  8. 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

  9. 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.

  10. Find the length of side a in the right triangle below. • It is not obvious which Pythagoreantriple 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.

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