The Converse of the Pythagorean Theorem

Geometry

Using the Converse• The Converse of the Pythagorean

Theorem is True.

• Remember “Converse” means “Reverse.”

Theorem 9.5: Converse of the Pythagorean Theorem

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

• If c2 = a2 + b2, then ∆ABC is a right triangle.

c

b

a

B

C A

Note:

• Use the Converse of the Pythagorean Theorem to show that a given triangle is a right triangle.

Ex. 1: Verifying Right Triangles

• Check to see whether the side lengths satisfy the equation

78

√113

The triangle is a right triangle.

36

15

Ex. 2: Verifying Right Triangles

4√95

The triangle is NOT a right triangle.

Classifying Triangles

• Sometimes it is hard to tell from looking at a triangle whether it is obtuse or acute. The theorems on the following slides can help you tell.

Theorem 9.6—Triangle Inequality

• 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 acute.

• If c2 < a2 + b2, then ∆ABC is acute

cb

a

A

C B

c2 < a2 + b2

Theorem 9.7—Triangle Inequality

• 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.

• If c2 > a2 + b2, then ∆ABC is obtuse

c2 > a2 + b2

c

b

a

A

C B

Ex. 2: Classifying Triangles

• Decide whether the set of numbers can represent the side lengths of a triangle. If they can, classify the triangle as right, acute or obtuse.

a. 38, 77, 86 b. 10.5, 36.5, 37.5

Triangle Inequality to confirmExample 2a

c2 = a2 + b2

862 = 382 + 772

c2 is greater than a2 + b2

The triangle is obtuse

Triangle Inequality to confirmExample 2bc2 = a2 + b2

37.52 = 10.52 + 36.52

c2 is less than a2 + b2

The triangle is acute

Ex. 3: Building a foundation

• Construction: You use four stakes and string to mark the foundation of a house. You want to make sure the foundation is rectangular.

A friend measures the four sides to be 30 feet, 30 feet, 72 feet, and 72 feet. He says these measurements prove that the foundation is rectangular. Is he correct?

Ex. 3: Building a foundation

• Solution: Your friend is not correct. The foundation could be a nonrectangular parallelogram, as shown below.

Ex. 3: Building a foundation

b. You measure one of the diagonals to be 78 feet. Explain how you can use this measurement to tell whether the foundation will be rectangular.

Ex. 3: Building a foundationSolution: The diagonal

divides the foundation into two triangles. Compare the square of the length of the longest side with the sum of the squares of the shorter sides of one of these triangles.

• Because 302 + 722 = 782, you can conclude that both the triangles are right triangles. The foundation is a parallelogram with two right angles, which implies that it is rectangular