Geometry: Pythagorean Theorem and Its Converse

In Figure 1 , CD is the altitude to hypotenuse AB.

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 a² = cx
and c/ b = b/ y, which becomes b² = 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 (leg² + leg² = hypotenuse²). See Figure 2 for the parts of a right triangle.

Figure 2

Parts of a right triangle.

Example 1: In Figure 3 , find 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 to find 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 a² + b² = c² 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 to find 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 to find x.

Figure 6

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

Subtract x² + 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 c² = a² + b², 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.