9.3 The Converse of the Pythagorean Theorem - PowerPoint PPT Presentation

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9.3 The Converse of the Pythagorean Theorem

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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. – PowerPoint PPT presentation

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Title: 9.3 The Converse of the Pythagorean Theorem


1
9.3 The Converse of the Pythagorean Theorem
  • Geometry
  • Ms. Reser

2
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.

3
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.

4
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.

5
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.

6
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
7
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.
8
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.
9
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.

10
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
11
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
12
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.

13
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

14
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

15
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?

16
Ex. 3 Building a foundation
  • Solution Your friend is not correct. The
    foundation could be a nonrectangular
    parallelogram, as shown below.

17
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.

18
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.

19
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.
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