Chapter 6.1: Similarity

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Ratios, Proportions, and the Geometric Mean

• Chapter 6.1 Similarity

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Ratios

• A ratio is a comparison of two numbers expressed
  by a fraction.
• The ratio of a to b can be written 3 ways
• ab
• a to b

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Equivalent Ratios

• Equivalent ratios are ratios that have the same
  value.
• Examples
• 12 and 36
• 515 and 13
• 636 and 16
• 218 and 19
• 416 and 14
• 735 and 15
• Can you come up with your own?

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Simplify the ratios to determine an equivalent
ratio.
3 ft 1 yard
Convert 3 yd to ft
1 km 1000 m
Convert 5 km to m
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Simplify the ratio
Convert 2 ft to in
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What is the simplified ratio of width to length?
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What is the simplified ratio of width to length?
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What is the simplified ratio of width to length?
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Use the number line to find the ratio of the
distances
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Finding side lengths with ratios and perimeters

• A rectangle has a perimeter of 56 and the ratio
  of length to width is 61.
• The length must be a multiple of 6, while the
  width must be a multiple of 1.
• New Ratio 6x1x, where 6x length and
  1x width
• What next?
• Length 6x, width 1x, perimeter 56
• 562(6x)2(1x)
• 5612x2x
• 5614x
• 4x
• L 24, w 4

P2l2w
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Finding side lengths with ratios and area

• A rectangle has an area of 525 and the ratio of
  length to width is 73
• A l²w
• Length 7x
• Width 3x
• Area 525
• 525 7x²3x
• 525 21x²
• v25 vx²
• 5 x

Length 7x 7(5) 35
Width 3x 3(5) 15
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Triangles and ratios finding interior angles

• The ratio of the 3 angles in a triangle are
  represented by 123.
• The 1st angle is a multiple of 1, the 2nd a
  multiple of 2 and the 3rd a multiple of 3.
• Angle 1 1x
• Angle 2 2x
• Angle 3 3x
• What do we know about the sum of the interior
  angles?

30 2(30) 60 3(30) 90
1x 2x 3x 180 6x 180 X 30
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Triangles and ratios finding interior angles

• The ratio of the angles in a triangle are
  represented by 112.
• Angle 1 1x
• Angle 2 1x
• Angle 3 2x
• 1x 1x 2x 180
• 4x 180
• x 45

Angle 1 1x 1(45) 45
Angle 2 1x 1(45) 45
Angle 3 2x 2(45) 90
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Proportions, extremes, means

• Proportion a mathematical statement that states
  that 2 ratios are equal to each other.

means
extremes
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Solving Proportions

• When you have 2 proportions or fractions that are
  set equal to each other, you can use cross
  multiplication.
• 1y 3(3)
• y 9

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Solving Proportions
1(8) 2x
4(15) 12z
8 2x
60 12z
4 x
5 z
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A little trickier
3(8) 6(x 3)
24 6x 18
42 6x
7 x
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Xs on both sides?
3(x 8) 6x
3x 24 6x
24 3x
8 x
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Now you try!
z 3
x 18
d 5
x 9
m 7
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Geometric Mean

• When given 2 positive numbers, a and b the
  geometric mean satisfies

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Find the geometric mean
x 2
x 3
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Find the geometric mean
x 9