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9.3 The Converse of the Pythagorean Theorem

- Geometry
- Ms. Reser

Objectives/Assignment

- Use the Converse of the Pythagorean Theorem to

solve problems. - Use side lengths to classify triangles by their

angle measures. - Assignment pp. 545-547 1-35
- Assignment due today 9.2
- Reminder Quiz after this section on Monday.

Using the Converse

- In Lesson 9.2, you learned that if a triangle is

a right triangle, then the square of the length

of the hypotenuse is equal to the sum of the

squares of the length of the legs. The Converse

of the Pythagorean Theorem is also true, as

stated on the following slide.

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.

Note

- You can use the Converse of the Pythagorean

Theorem to verify that a given triangle is a

right triangle, as shown in Example 1.

Ex. 1 Verifying Right Triangles

- The triangles on the slides that follow appear to

be right triangles. Tell whether they are right

triangles or not.

v113

4v95

Ex. 1a Verifying Right Triangles

- Let c represent the length of the longest side of

the triangle. Check to see whether the side

lengths satisfy the equation c2 a2 b2. - (v113)2 72 82
- 113 49 64
- 113 113 ?

v113

?

?

The triangle is a right triangle.

Ex. 1b Verifying Right Triangles

- c2 a2 b2.
- (4v95)2 152 362
- 42 (v95)2 152 362
- 16 95 2251296
- 1520 ? 1521 ?

4v95

?

?

?

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 9.6Triangle 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 lt a2 b2, then ?ABC is acute

c2 lt a2 b2

Theorem 9.7Triangle 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 gt a2 b2, then ?ABC is obtuse

c2 gt a2 b2

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. - 38, 77, 86 b. 10.5, 36.5, 37.5
- You can use the Triangle Inequality to confirm

that each set of numbers can represent the side

lengths of a triangle. Compare the square o the

length of the longest side with the sum of the

squares of the two shorter sides.

Triangle Inequality to confirm Example 2a

- Statement
- c2 ? a2 b2
- 862 ? 382 772
- 7396 ? 1444 5959
- 7395 gt 7373

- Reason
- Compare c2 with a2 b2
- Substitute values
- Multiply
- c2 is greater than a2 b2
- The triangle is obtuse

Triangle Inequality to confirm Example 2b

- Statement
- c2 ? a2 b2
- 37.52 ? 10.52 36.52
- 1406.25 ? 110.25 1332.25
- 1406.24 lt 1442.5

- Reason
- Compare c2 with a2 b2
- Substitute values
- Multiply
- 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. 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 foundation

- 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

- Solution 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.

Reminders

- Monday, March 14 Quiz over 9.1-9.3
- Thursday, March 17 Quiz over 9.4-9.5
- Thursday, March 24 Chapter 9 Test/Binder Check

If you plan on leaving earlier than Spring

BreakDo not forget to take your chapter 9 exam.

It would be in your best interest to get it out

of the way.