In Figure 1

**Figure 1 **An altitude drawn to the hypotenuse of a right triangle to aid in deriving the *Pythagorean theorem*.

So, by *Theorem 63*,

*c*/*a*=*a*/*x*, which becomes*a*^{2}=*cx*

- and
*c*/*b*=*b*/*y*, which becomes*b*^{2}=*cy*

From the addition property of equations in **algebra**, we get the following equation.

By factoring out the *c* on the right side,

But *x* + *y* = *c* *(Segment Addition Postulate)*,

This result is known as the *Pythagorean Theorem*.

*Theorem 65 (Pythagorean Theorem):* In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse (leg^{2} + leg^{2} = hypotenuse^{2}). See Figure 2

**Figure 2** Parts of a right triangle.

**Example 1:** In Figure 3*x*, the length of the hypotenuse.

**Figure 3 **Using the *Pythagorean Theorem* to find the hypotenuse of a right triangle.

**Example 2:** Use Figure 4 *x*.

**Figure 4 **Using the *Pythagorean Theorem* to find the hypotenuse of a right triangle.

Any three natural numbers, *a, b, c*, that make the sentence *a*^{2} + *b*^{2} = *c*^{2} true are called a Pythagorean triple. Therefore, 3‐4‐5 is called a Pythagorean triple. Some other values for *a*, *b*, and *c* that will work are 5‐12‐13 and 8‐15‐17. Any multiple of one of these triples will also work. For example, using the 3‐4‐5: 6‐8‐10, 9‐12‐15, and 15‐20‐25 are also Pythagorean triples.

**Example 3:** Use Figure 5 *x*.

**Figure 5 **Using the *Pythagorean Theorem* to find a leg of a right triangle.

If you can recognize that the numbers *x*, 24, 26 are a multiple of the 5‐12‐13 Pythagorean triple, the answer for *x* is quickly found. Because 24 = 2(12) and 26 = 2(13), then *x* = 2(5) or *x* = 10. You can also find *x* by using the *Pythagorean Theorem*.

**Example 4:** Use Figure 6 *x*.

**Figure 6** Using the *Pythagorean Theorem* to find the unknown parts of a right triangle.

Subtract *x*^{2} + 12 *x* + 36 from both sides.

But *x* is a length, so it cannot be negative. Therefore, *x* = 9.

The converse (reverse) of the *Pythagorean Theorem* is also true.

*Theorem 66:* If a triangle has sides of lengths *a, b,* and *c* where *c* is the longest length and *c*^{2} = *a*^{2} + *b*^{2}, then the triangle is a right triangle with *c* its hypotenuse.

**Example 5:** Determine if the following sets of lengths could be the sides of a right triangle: (a) 6‐5‐4, (b) , (c) 3/4‐1‐5/4.

(a) Because 6 is the longest length, do the following check.

So 4‐5‐6 are not the sides of a right triangle.

(b) Because 5 is the longest length, do the following check.

So are sides of a right triangle, and 5 is the length of the hypotenuse.

(c) Because 5/4 is the longest length, do the following check.

So 3/4‐1‐5/4 are sides of a right triangle, and 5/4 is the length of the hypotenuse.