Transcript answer

Chapter 6
Similarity
Pre-Requisite Skills
Page 354 all
6.1 Ratios, Proportions, and
the Geometric Mean
Objective: Solve problems by
writing and solving proportions
Ratio
• Two numbers or quantities being
compared
a to b
a:b
a/b
Simplifying Ratios
• Just like simplifying a fraction
• 72 : 9
EXAMPLE 1
Simplify ratios
Simplify the ratio.
a. 64 m : 6 m
SOLUTION
a.
Write 64 m : 6 m as 64 m .
6m
Then divide out the units and simplify.
64 m = 32 = 32 : 3
6m
3
GUIDED PRACTICE
Simplify the ratio.
1.
24 yards to 3 yards
ANSWER 8 : 1
for Example 1
Simplifying Ratios with Different
Units
• What to do:
Use unit analysis to cancel out units
*note: simplifying the numbers can be done
before or after canceling units
Example: Simplify 36in : 9ft
Examples
b.
5 ft
20 in.
2.
150 cm : 6 m
ANSWER 1 : 4
Reading Ratios
• The teacher to student ratio is 1 to 22.
• What does this mean?
Using Ratios and to Solve
Problems
Example
• You are planning to paint a mural on a
rectangular wall. You know the perimeter
of the wall is 484 feet. The ratio of its
length to its width is 9:2. Find the area of
the wall.
Guided Practice #3 page 357
The perimeter of a room is 48 feet and the
ratio of its length to its width is 7:5. Find
the length and width of the room.
Example
• The area of a rectangular garden is 108
square feet, and the ratio of the length to
the width is 4:3. Find the length and width
of fence needed to enclose the garden.
EXAMPLE 3
Use extended ratios
ALGEBRA The measures of the angles in
CDE are
in the extended ratio of 1 : 2 : 3. Find the measures of
the angles.
SOLUTION
Begin by sketching the triangle. Then use the
extended ratio of 1 : 2 : 3 to label the measures
as x° , 2x° , and 3x° .
o
o
o
o
Triangle Sum Theorem
x + 2x + 3x = 180
6x = 180
Combine like terms.
Divide each side by 6.
x = 30
ANSWER
o
o
o
o
o
The angle measures are 30 , 2(30 ) = 60 , and 3(30 ) = 90.
GUIDED PRACTICE
for Examples 2 and 3
4. A triangle’s angle measures are in the extended
ratio of 1 : 3 : 5. Find the measures of the angles.
ANSWER 20°, 60°, 100°
Proportions
• Ratios that are = to each other
• Means and Extremes (page 358)
EXAMPLE 4
Solve the proportion.
ALGEBRA
a.
Solve proportions
5 = x
10
16
SOLUTION
a.
5
10
x
16
Write original proportion.
5 16 =
10 x
Cross Products Property
80
10 x
Multiply.
x
Divide each side by 10.
=
=
8 =
EXAMPLE 4
b.
Solve proportions
2
1
=
y+1
3y
SOLUTION
b.
1
y+1 =
2
3y
Write original proportion.
1 3y
=
2 (y + 1)
Cross Products Property
3y
=
2y + 2
Distributive Property
y
=
2
Subtract 2y from each side.
GUIDED PRACTICE
Solve the proportion.
5. 2 = 5
x
8
16
ANSWER
5
1
4
6.
=
x–3
3x
ANSWER
7.
12
y–3
y
7 = 14
ANSWER
6
for Example 4
EXAMPLE 5
Solve a real-world problem
SCIENCE
As part of an environmental study, you need to
estimate the number of trees in a 150 acre area. You
count 270 trees in a 2 acre area and you notice that the
trees seem to be evenly distributed. Estimate the total
number of trees.
SOLUTION
Write and solve a proportion involving two ratios that
compare the number of trees with the area of the land.
EXAMPLE 5
Solve a real-world problem
270
n
=
2
150
270 150 = 2
20,250 = n
number of trees Write proportion.
area in acres
n
Cross Products Property
Simplify.
There are about 20,250 trees in the 150 acre area.
GUIDED PRACTICE
8.
for Examples 5 and 6
WHAT IF ? In Example 5, suppose you count 390
trees in a 3 acre area of the 150 acre area. Make
a new estimate of the total number of trees.
ANSWER
19,500 trees
Geometric Mean
• A number x that satisfies the proportion
a/x = x/b
• To find the geometric mean of two
numbers, multiply them together and take
the square root
– Then simplify the square root (no decimal
answers)
EXAMPLE 6
Find a geometric mean
Find the geometric mean of 24 and 48.
SOLUTION
x =
ab
Definition of geometric mean
=
24 48
Substitute 24 for a and 48 for b.
=
24 24 2
Factor.
= 24
2
Simplify.
The geometric mean of 24 and 48 is 24
2
33.9.
GUIDED PRACTICE
for Examples 5 and 6
Find the geometric mean of the two numbers.
9.
12 and 27
ANSWER
10.
18 and 54
ANSWER
11.
18
18
3
16 and 18
ANSWER
12
2
Daily Homework Quiz
1.
Simplify the ratio 1200 cm : 1.8 m.
ANSWER
2.
For use after Lesson 6.1
2:3
The perimeter of a rectangle is 528 millimeters .
The ratio of length of the width is 8 : 3. Find the
length and the width.
ANSWER
192 mm, 72 mm
Daily Homework Quiz
3.
Solve
ANSWER
4.
For use after Lesson 6.1
1 = 3
s+8
36
4
Find the geometric mean of 42 and 12
ANSWER
6 14
Daily Homework Quiz
5.
The extended ratio of the angles of a triangle is
5: 12: 13. Find the angle measures of the triangle.
ANSWER
6.
For use after Lesson 6.1
o
o
o
30 ,72 , 78
The area of a rectangle is 720 square inches. If the
ratio of the length to the width is 5 : 4, find the
perimeter of the rectangle.
ANSWER
108 in.
Homework
• 1, 2 – 52 evens, 58 – 66 evens