Chapter 8:Right Triangles

and TrigonometrySection 8-1:

The Pythagorean Theorem

and its Converse

Objectives:

To use the Pythagorean Theorem.

To use the converse of the Pythagorean Theorem.

Vocabulary

Pythagorean Triple

Pythagoras

Greek mathematician from the 6th century BC.

Famous for the Pythagorean Theorem

Others knew of the Pythagorean Theorem first:

Babylonians

Egyptians

Chinese

Theorem 8-1:“The Pythagorean Theorem”

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.

2 2 2a b c

Pythagorean Triple

A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation:

2 2 2a b c

Example

Solve for the variable. Do the sides of the triangle form a Pythagorean triple?

21

20x

Example

Solve for the variable. Do the sides of the triangle form a Pythagorean triple?

1634

y

Example

Solve for the variable. Do the sides of the triangle form a Pythagorean triple?

4

8z

Theorem 8-2:“Converse of the Pythagorean Theorem”

If the square of the lengths of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

Using the Converse of the Pythagorean Theorem

Is the triangle a right triangle?

6

108

Using the Converse of the Pythagorean Theorem

Is the triangle a right triangle?

6

2

5

*Note:If a triangle is not a right triangle, then it is

either an acute triangle or an obtuse triangle.

Theorem 8-3:

If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

2 2 2If , then the triangle is obtuse.c a b

Theorem 8-4:

If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

2 2 2If , then the triangle is acute.c a b

Classify the Triangles asAcute, Obtuse, or Right.

7, 8, and 11

16, 19, and 24

5, 7, and 10