Pythagorean Theorem

Section 11.4

P. 737

• Pythagoras was born around 570 BC on the island of Samos in ancient Greece.

• That is about 2,570 years ago!

Pythagoras

• He founded a school in southern Italy after traveling in Egypt and the Middle East. He was a philosopher, musician, and astronomer, but he is most remembered as a mathematician.

• The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90o angle. As a matter of fact, they had a rope with 12 evenly spaced knots like this one:

•

• that they used to cut stones and build perfect corners in their buildings and pyramids. It is believed that they only knew about the 3, 4, 5 triangle and not the general theorem that applies to all right triangles.

• In any right triangle, the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.

• The Pythagorean Theorem is one of the most important theorems in the whole realm of geometry.

• When the two shorter sides in a right triangle are squared and then added, the sum equals the square of the longest side or hypotenuse.

The Pythagorean TheoremWhere c is the length of the hypotenuse and

a and b are the lengths of the other two sides, the theorem can be expressed as the following equation:

• c2 – a2 = b2

• c2 – b2 = a2

• The Pythagorean Theorem

• If a triangle is a right triangle, then the sum of the squares of the lengths of the legs a and b equals the square of the length of the hypotenuse c.

• a2 + b2 = c2a

c

b

• Also true is:

c2 - b2 = a2

• This would be used to find one of the legs if you know the length of the hypotenuse and the other leg.

EXAMPLE 1 Use the Pythagorean theorem

Find the unknown length for the triangle shown.

SOLUTION

Pythagorean theorem

Substitute 6 for b and 7 for c.

Simplify.

Subtract 36 from each side.

Take positive square root of each side.

a + b = c2 2 2

a + 6 = 72 2 2

a + 36 = 492

a = 132

a = 13

The side length a is = 13ANSWER

GUIDED PRACTICE for Example 1

1. The lengths of the legs of a right triangle are a = 5 and b = 12. Find c.

c = 13ANSWER

• a2 + b2 = c2 a & b = legs, c = hypot• a = 6, b = 8 c = ____• a = 6 b = 6 c = _____• a = 10 b = 24 c = _____• a = 2 c = 7 b = _____

• Determine if the given lengths are sides of a right triangle.

• 5, 11, 12

• 20, 29, 21

• 11.9, 12, 16.9

• Assignment: P. 740• 1-15, 35

• Make sure you put formula / work on your paper!

• Round to the nearest tenths.

P. 740 1-15, 35