9.3 The Converse of the Pythagorean Theorem

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9.3 The Converse of the Pythagorean Theorem

• Geometry
• Ms. Reser

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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.

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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.

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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.

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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.

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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
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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.
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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.
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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.

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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
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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
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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.

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Triangle Inequality to confirmExample 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

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Triangle Inequality to confirmExample 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

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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?

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Ex. 3 Building a foundation

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

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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.

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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.

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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.