Pythagorean Theorem

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Pythagorean Theorem
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VOCABULARY

• Right Triangle
• Hypotenuse
• Legs
• Pythagorean Theorem
• Radical/Radicand
• Square Root

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RIGHT TRIANGLE

• Longest side is the hypotenuse, side c (opposite
  the 90o angle)
• The other two sides are the legs, sides a and b
• Pythagoras developed a formula for finding the
  length of the sides of any right triangle

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RADICAL/RADICAND
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SQUARE ROOT

• Definition A number the produces a specified
  quantity when multiplied by itself.

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THE PYTHAGOREAN THEOREM

• For any right triangle, the sum of the areas of
  the two small squares is equal to the area of the
  larger.
• a2 b2 c2

• In different words
• If the angle opposite the hypotenuse is a right
  angle, then a2 b2 c2

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EXAMPLE 1

• If two measures of the sides of a right triangle
  are known, the Pythagorean Theorem can be used to
  find the measure of the third side.

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EXAMPLE 2

• 15282c2
• 22564c2
• 289c2
• 17c

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YOUR TURN! ?
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CONVERSE?

• Normal If x, then y.
• Converse If y, then x.
• Example
• If it is raining, then the grass is wet.
• Converse If the grass is wet, then it is
  raining.

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CONVERSE OF THE PYTHAGOREAN THEOREM

• If a triangle has sides of lengths a, b, and c
  and a2 b2 c2
• then the triangle is a right triangle with
  hypotenuse of length c.

You can determine if a triangle is a right
triangle if the 3 side lengths fit into a2 b2
c2
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EXAMPLE 1
No! This is not a right triangle.

• 527292
• 254981?

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YOUR TURN! ?
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COMMON PYTHAGOREAN TRIPLES
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Find the length of the hypotenuse if1. a 12
and b 16.

• 122 162 c2
• 144 256 c2
• 400 c2
• Take the square root of both sides.

20 c
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Find the length of the hypotenuse if2. a 5
and b 7.
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Find the length of the hypotenuse given a 6 and
b 12

1. 180
2. 324
3. 13.42
4. 18

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Find the length of the leg, to the nearest
hundredth, if3. a 4 and c 10.
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Find the length of the leg, to the nearest
hundredth, if4. c 10 and b 7.
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Find the length of the missing side given a 4
and c 5

1. 1
2. 3
3. 6.4
4. 9

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5. The measures of three sides of a triangle are
given below. Determine whether each triangle is
a right triangle. 5 , 3, and 8
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The rectangular bottom of a box has an interior
length of 19 inches and an interior width of 14
inches. A stick is placed in the box along the
diagonal of the bottom of the box. Which
measurement is closest to the longest possible
length for the stick? a2 b2 c2 F 12.8
in. G 16.3 in. H 23.6 in. J 33.0 in.
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REAL-LIFE APPLICATIONS

• The Pythagorean theorem has far-reaching
  ramifications in other fields (such as the arts),
  as well as practical applications.
• The theorem is invaluable when computing
  distances between two points, such as in
  navigation and land surveying.
• Another important application is in the design of
  ramps. Ramp designs for handicap-accessible sites
  and for skateboard parks are very much in demand.
  
• Can you think of other ways the Pythagorean
  Theorem can be extremely valuable?

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Baseball Problem

• A baseball diamond is really a square.
• You can use the Pythagorean theorem to find
  distances around a baseball diamond.

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BASEBALL PROBLEM

• The distance between
• consecutive bases is 90
• feet. How far does a
• catcher have to throw
• the ball from home
• plate to second base?

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BASEBALL PROBLEM

• To use the Pythagorean theorem to solve for x,
  find the right angle.
• Which side is the hypotenuse?
• Which sides are the legs?
• Now use a2 b2 c2

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BASEBALL PROBLEMSOLUTION

• The hypotenuse is the distance from home to
  second, or side x in the picture.
• The legs are from home to first and from first to
  second.
• Solution
• x2 902 902 16,200
• x 127.28 ft

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LADDER PROBLEM

• A ladder leans against a second-story window of a
  house. If the ladder is 25 meters long, and the
  base of the ladder is 7 meters from the house,
  how high is the window?

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LADDER PROBLEMSOLUTION

• First draw a diagram that shows the sides of the
  right triangle.
• Label the sides
• Ladder is 25 m
• Distance from house is 7 m
• Use a2 b2 c2 to solve for the missing side.

Distance from house 7 meters
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LADDER PROBLEMSOLUTION

• 72 b2 252
• 49 b2 625
• b2 576
• b 24 m
• How did you do?

B
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DO NOT DO THIS!
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