Ratio, Proportion, and Triangle Applications

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Ratio, Proportion, and Triangle Applications
Chapter Six

• 6.1 Ratios Rates
• 6.2 Proportions
• 6.3 Proportions and Problem Solving
• 6.4 Square Roots the Pythagorean Theorem
• 6.5 Congruent Similar Triangles

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Ratio and Rates
Section 6.1
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Writing Ratios as Fractions
A ratio is the quotient of two numbers.
For example, a percent can be thought of as a
ratio, since it is the quotient of a number and
100.
or the ratio of 53 to 100
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Ratio
The ratio of a number a to a number b is their
quotient. Ways of writing ratios are
a to b,
a b,
and
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Writing Rates as Fractions
A rate is a special kind of ratio. It is used to
compare different kinds of quantities.
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Finding Unit Rates
To write a rate as a unit rate, divide the
numerator of the rate by the denominator.
314.5 17 18.5
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Finding Unit Prices
When a unit rate is money per item, it is also
called a unit price.
A store charges 2.76 for a 12-ounce jar of
pickles. What is the unit price?
(0.23 per ounce )
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Proportions
Section 6.2
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A proportion is a statement that two ratios or
rates are equal.
Solving Proportions
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Solving Proportions . . .
A proportion contains four numbers. If any three
numbers are known, the fourth number can be found
by solving the proportion. To solve use cross
products.
Multiply both sides by the LCD, bd
These are called cross products.
ad bc
Simplify
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Determining Whether Proportions are True
True proportion
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Finding Unknown Numbers in Proportions
Cross multiply.
Simplify the left side.
Divide both sides by 28.
Check
(Rounded)
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Proportions and Problem Solving
Section 6.3
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Solving Problems by Writing Proportions
A 16-oz Cinnamon Mocha Iced Tea at a local
coffee shop has 80 calories. How many calories
are there in a 28-oz Cinnamon Mocha Iced Tea?
Solve the proportion.
Cross multiply.
Simplify the right side.
Divide both side by 140.
A 28-oz Cinnamon Mocha Iced Tea has 140 calories.
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Helpful Hint
When writing proportions to solve problems, write
the proportions so that the numerators have the
same unit measures and the denominators have the
same unit measures.
For example,
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Square Roots and the Pythagorean Theorem
Section 6.4
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The square of a number is the number times itself.
The square of 6 is 36 because .
The square of - 6 is also 36 because
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The reverse process of squaring is finding a
square root.
A square root of 36 is 6 because .
A square root of 36 is also 6 because

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Square Root of a Number
0.
Also,
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Helpful Hint
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Numbers like are
called perfect squares because their square
root is a whole number or a fraction.
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Approximating Square Roots
A square root such as cannot be written as
a whole number or a fraction since 6 is not a
perfect square. It can be approximated by
estimating, by using a table, or by using a
calculator.
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One important application of square roots has to
do with right triangles.
A right triangle is a triangle in which one of
the angles is a right angle or measures 90º
(degrees).
The hypotenuse of a right triangle is the side
opposite the right angle.
The legs of a right triangle are the other two
sides.
hypotenuse
leg
leg
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Pythagorean Theorem
If a and b are the lengths of the legs of a
right triangle and c is the length of the
hypotenuse, then
In other words,
(leg)2 (other leg)2 (hypotenuse)2.
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Congruent and Similar Triangles
Section 6.5
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Congruent Triangles
Two triangles are congruent when they have the
same shape and the same size. Corresponding
angles are equal, and corresponding sides are
equal.
equal angles
a 6
c 11
d 6
e 11
b 9
f 9
equal angles
equal angles
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Similar Triangles
Similar triangles are found in art, engineering,
architecture, biology, and chemistry. Two
triangles are similar when they have the same
shape but not necessarily the same size.
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In similar triangles, the measures of
corresponding angles are equal and corresponding
sides are in proportion.
d 6
a 3
e 10
b 5
c 8
f 16
Side a corresponds to side d, side b corresponds
to side e, and side c corresponds to side f.
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