Pythagorean Theorem and Corollaries

Let a, b, and c be the lengths of the sides of ΔABC, with a ≤ b ≤ c.

• a² + b² = c² if and only if angle C is a right angle.
• a² + b² < c² if and only if angle C is obtuse.
• a² + b² > c² if and only if angle C is acute.

TIP

The Pythagorean theorem is probably the most important theorem you need to know. Be sure to review all of its uses.

EXAMPLE 4: Since 8² + 15² = 64 + 225 = 289 = 17², a triangle whose sides have lengths 8, 15, and 17 is a right triangle. Since 10² + 15² = 100 + 225 = 325 < 19², a triangle whose sides have lengths 10, 15, and 19 is not a right triangle—it is obtuse.

On the Math 1 test, the most common right triangles whose sides are integers are the 3-4-5 right triangle and its multiples.

Key Fact H6

For any positive number x, there is a right triangle whose sides are 3x, 4x, 5x.

For example:

x = 1	3, 4, 5	x = 5	15, 20, 25
x = 2	6, 8, 10	x = 10	30, 40, 50
x = 3	9, 12, 15	x = 50	150, 200, 250
x = 4	12, 16, 20	x = 100	300, 400, 500

NOTE

KEY FACT H6 applies even if x is not an integer. For example:

x = 0.5 1.5, 2, 2.5

x = π  3π, 4π, 5π

Other right triangles with integer sides that you should recognize immediately are the ones whose sides are 5, 12, 13 and 8, 15, 17. These sets of three numbers are often referred to as Pythagorean triples.