Ratios, Proportions and Similar Figures

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Ratios, Proportions and Similar Figures

• Ratios, proportions and scale drawings

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There are many uses of ratios and proportions.
We use them in map reading, making scale drawings
and models, solving problems.
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The most recognizable use of ratios and
proportions is drawing models and plans for
construction. Scales must be used to approximate
what the actual object will be like.
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A ratio is a comparison of two quantities by
division. In the rectangles below, the ratio of
shaded area to unshaded area is 12, 24, 36,
and 48. All the rectangles have equivalent
shaded areas. Ratios that make the same
comparison are equivalent ratios.
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Using ratiosThe ratio of faculty members to
students in one school is 115. There are 675
students. How many faculty members are
there?faculty 1students 151
x15 67515x 675 x
45 faculty

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A ratio of one number to another number is the
quotient of the first number divided by the
second. (As long as the second number ? 0)
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A ratio can be written in a variety of ways.

• You can use ratios to compare quantities or
  describe rates. Proportions are used in many
  fields, including construction, photography, and
  medicine.

ab a/b a to b
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Since ratios that make the same comparison are
equivalent ratios, they all reduce to the same
value.
2 3 1 10 15 5


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Proportions

• Two ratios that are equal

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A proportion is an equation that states that two
ratios are equal, such as
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In simple proportions, all you need to do is
examine the fractions. If the fractions both
reduce to the same value, the proportion is
true.This is a true proportion, since both
fractions reduce to 1/3.
5 2 15 6

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In simple proportions, you can use this same
approach when solving for a missing part of a
proportion. Remember that both fractions must
reduce to the same value.

• To determine the unknown value you must cross
  multiply. (3)(x) (2)(9)
• 3x 18
• x 6
• Check your proportion
• (3)(x) (2)(9)
• (3)(6) (2)(9)
• 18 18 True!

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So, ratios that are equivalent are said to be
proportional. Cross Multiply makes solving or
proving proportions much easier. In this example
3x 18, x 6.

• If you remember, this is like finding equivalent
  fractions when you are adding or subtracting
  fractions.

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1) Are the following true proportions?
2 10 3 15
2 10 3 5


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2) Solve for x
4 x 6 42

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3) Solve for x
25 5 x 2

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Solve the following problems.

• 4) If 4 tickets to a show cost 9.00, find the
  cost of 14 tickets.

• 5) A house which is appraised for 10,000 pays
  300 in taxes. What should the tax be on a house
  appraised at 15,000.

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Similar Figures

• The Big and Small of it

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For Polygons to be Similarcorresponding angles
must be congruent, andcorresponding sides must
be proportional(in other words the sides must
have lengths that form equivalent ratios)
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Congruent figures have the same size and shape.
Similar figures have the same shape but not
necessarily the same size. The two figures below
are similar. They have the same shape but not
the same size.
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Lets look at the two triangles we looked at
earlier to see if they are similar.Are the
corresponding angles in the two triangles
congruent?Are the corresponding sides
proportional? (Do they form equivalent ratios)
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Just as we solved for variables in earlier
proportions, we can solve for variables to find
unknown sides in similar figures.Set up the
corresponding sides as a proportion and then
solve for x.
Ratios x/12 and 5/10
x 5 12 10 10x 60 x 6
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Determine if the two triangles are similar.
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In the diagram we can use proportions to
determine the height of the tree.5/x 8/288x
140x 17.5 ft
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The two windows below are similar. Find the
unknown width of the larger window.
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These two buildings are similar. Find the height
of the large building.