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
= 52
= 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
= 132
= 169
169 = 169

7. Show that {2, 2, 5} is not a Pythagorean triple.
a2 + b2 = c2
= 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
= 102
= 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 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.