12'3 The Pythagorean Theorem

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12.3 The Pythagorean Theorem

• CORD
• Mrs. Spitz
• Spring 2007

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Objectives/Assignment

• Use Pythagorean Theorem
• Assignment pp. 484-485 4-39 all
• Assignment due today 12.2 p. 479-480 5-51

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History Lesson

• Around the 6th century BC, the Greek
  mathematician Pythagorus founded a school for the
  study of philosophy, mathematics and science.
  Many people believe that an early proof of the
  Pythagorean Theorem came from this school.
• Today, the Pythagorean Theorem is one of the most
  famous theorems in geometry. Over 100 different
  proofs now exist.

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Proving the Pythagorean Theorem

• In this lesson, you will study one of the most
  famous theorems in mathematicsthe Pythagorean
  Theorem. The relationship it describes has been
  known for thousands of years.

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Theorem 9.4 Pythagorean Theorem

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

c2 a2 b2
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Using the Pythagorean Theorem

• A Pythagorean triple is a set of three positive
  integers a, b, and c that satisfy the equation c2
  a2 b2 For example, the integers 3, 4 and 5
  form a Pythagorean Triple because 52 32 42.

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Ex. 1 Finding the length of the hypotenuse.

• Find the length of the hypotenuse of the right
  triangle. Tell whether the sides lengths form a
  Pythagorean Triple.

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Solution

• (hypotenuse)2 (leg)2 (leg)2
• x2 52 122
• x2 25 144
• x2 169
• x 13
• Because the side lengths 5, 12 and 13 are
  integers, they form a Pythagorean Triple. Many
  right triangles have side lengths that do not
  form a Pythagorean Triple as shown next slide.

• Pythagorean Theorem
• Substitute values.
• Multiply
• Add
• Find the positive square root.
• Note There are no negative square roots until
  you get to Algebra II and introduced to
  imaginary numbers.

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Ex. 2 Finding the Length of a Leg

• Find the length of the leg of the right triangle.

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Solution

• (hypotenuse)2 (leg)2 (leg)2
• 142 72 x2
• 196 49 x2
• 147 x2
• v147 x
• v49 v3 x
• 7v3 x

• Pythagorean Theorem
• Substitute values.
• Multiply
• Subtract 49 from each side
• Find the positive square root.
• Use Product property
• Simplify the radical.

• In example 2, the side length was written as a
  radical in the simplest form. In real-life
  problems, it is often more convenient to use a
  calculator to write a decimal approximation of
  the side length. For instance, in Example 2, x
  7 v3 12.1

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Note

• Determine if the following lengths can represent
  the sides of a right triangle.

Right ?
c2 a2 b2
Acute ?
c2 lt a2 b2
c2 gt a2 b2
Obtuse ?
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32. 12, 11, 15

• The measures of the sides of a triangle are
  given. Determine whether each triangle is a
  right triangle.
• c2 a2 b2
• 152 112 122
• 225 121 144?
• 225 ? 265 Not a right Triangle

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14 a v7 b v9 c?

• c2 a2 b2
• c2 v72 v92
• c2 7 9
• c2 16
• c 4

• Note

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Area of a rectangle

• The area of a rectangle is 40 square meters.
  Find the length of a diagonal of the rectangle if
  its length is 2 meters less than twice its width.

?
40 bh
b 2h 2
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Ex. 3 Finding the area of a triangle

• Find the area of the triangle to the nearest
  tenth of a meter.
• You are given that the base of the triangle is 10
  meters, but you do not know the height.

Because the triangle is isosceles, it can be
divided into two congruent triangles with the
given dimensions. Use the Pythagorean Theorem to
find the value of h.
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Solution

• Steps
• (hypotenuse)2 (leg)2 (leg)2
• 72 52 h2
• 49 25 h2
• 24 h2
• v24 h

• Reason
• Pythagorean Theorem
• Substitute values.
• Multiply
• Subtract 25 both sides
• Find the positive square root.

Now find the area of the original triangle.
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Area of a Triangle

• Area ½ bh
• ½ (10)(v24)
• 24.5 m2
• The area of the triangle is about 24.5 m2