Pythagorean Theorem

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Pythagorean Theorem

• Pre-Algebra

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Lesson Topics

• Baseball
• Definitions
• Pythagorean Theorem
• Converse of the Pythagorean Theorem
• Application of the Pythagorean Theorem

Pythagoras
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Baseball

• A baseball scout uses many different tests
  to determine whether or not to draft a particular
  player. One test for catchers is to see how
  quickly they can throw a ball from home plate to
  second base. The scout must know the distance
  between the two bases in case a player cannot be
  tested on a baseball diamond. This distance can
  be found by separating the baseball diamond into
  two right triangles.

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Right Triangles

• Right Triangle A triangle with one right angle.
• Hypotenuse Side opposite the right angle and
  longest side of a right triangle.
• Leg Either of the two sides that form the right
  angle.

Leg
Hypotenuse
Leg
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Pythagorean Theorem

• In a right triangle, if a and b are the measures
  of the legs and c is the measure of the
  hypotenuse, then
• a2 b2 c2.
• This theorem is used to find the length of any
  right triangle when the lengths of the other two
  sides are known.

c
b
a
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Finding the Hypotenuse
a2 b2 c2

• Example 1 Find the length of the hypotenuse of
  a right triangle if a 3 and
• b 4.

4
3
c
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Finding the Length of a Leg

• Example 2 Find the length of the leg of the
  following right triangle.

a2 b2 c2
__________________
12
a
9
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Examples of the Pythagorean Theorem

• Example 3 Find the length of the hypotenuse c
  when a 11 and b 4.
  Solution

• Example 4 Find the length of the leg of the
  following right triangle.
• Solution
  

13
c
11
a
4
5
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Solution of Example 3

• Find the length of the hypotenuse c when
• a 11 and b 4.

• a2 b2 c2

c
11
4
10
Solution of Example 4

• Example 4 Find the length of the leg of the
  following right triangle.

_______________
13
a
5
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Converse of the Pythagorean Theorem

• If a2 b2 c2, then the triangle with sides a,
  b, and c is a right triangle.
• If a, b, and c satisfy the equation
• a2 b2 c2, then a, b, and c are known as
  Pythagorean triples.

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Example of the Converse

• Example 5 Determine whether a triangle with
  lengths 7, 11, and 12 form a right triangle.
• The hypotenuse is the longest length.

This is not a right triangle.
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Example of the Converse

• Example 6 Determine whether a triangle with
  lengths 12, 16, and 20 form a right triangle.

This is a right triangle. A set of integers such
as 12, 16, and 20 is a Pythagorean triple.
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Converse Examples

• Example 7 Determine whether 4, 5, 6 is a
  Pythagorean triple.
• Example 8 Determine whether 15, 8, and 17 is a
  Pythagorean triple.

4, 5, and 6 is not a Pythagorean triple.
15, 8, and 17 is a Pythagorean triple.
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Baseball Problem

• On a baseball diamond, the hypotenuse is the
  length from home plate to second plate. The
  distance from one base to the next is 90 feet.
  The Pythagorean theorem can be used to find the
  distance between home plate to second base.

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Solution to Baseball Problem

• For the baseball diamond, a 90 and
• b 90.

c
90
The distance from home plate to second base is
approximately 127 feet.
90
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Homework

• Worksheet due tomorrow