Ch 6.1
• Ratio- comparison of two quantities
– Ex:
3
4
or
3:4
• Ratios are UNITLESS
– Means without units ( or units cancel out)
• A ratio in which the denominator is 1 is called
a unit ratio
Proportion
• Proportion – an equation stating that two
ratios are equal
– Equivalent fractions set equal to each other form
proportions
– Ex:
2 = 6
3
9
(equivalent fractions are fractions that reduce to the
same quantity)
Solve
The U.S. Census Bureau surveyed 8,218 schools
nationally about their girls’ soccer program.
They found that 270,273 girls participated in a
high school soccer program in the 1999-2000
school year. Find the ratio of girl soccer players
per school to the nearest tenth.
• Girls = 270,273 = 32.8879 = 32.9
schools 8,218
Solve
In a triangle, the ratio of the measures of three
sides is 4:6:9, and its perimeter is 190 inches.
Find the longest side of the triangle.
4x + 6x + 9x = 190
19x = 190
X = 10
4x = 40 6x = 60 9x = 90
Pg 285
5) A replica of The Thinker is 10 inches tall. A statue of The Thinker, located in
front of Grawemeyer Hall on the Belnap Campus of the University of Louisville
in Kentucky, is 10 ft tall. What is the ratio of the replica to the statue in
Louisville?
10 in. replica  covert inches to ft with ratio 1ft
10 ft statue
12 in
* 1 foot has 12 inches
10in * 1ft = 10ft
12in 12
1:12
1/12
10ft
12 . = 10ft * 1
= 1/12
10ft
12 10 ft
The replica is 1/12 the size of the statue
Cross Products
• Every proportion has cross products
– Ex:
2 = 6
3
9
Extremes (2) (9) = Means (3) (6)
18 = 18
The product of the means equals the product of
the extremes, so the cross products are equal.
Solve
1. 3 = x
5
75
2. 3x -5 = -13
4
2
Pg 286 YOU SOLVE
38. The ratios of the
lengths of the sides of
three polygons are given
below. Make a conjecture
about identifying each
type of polygon.
a. 2:2:3
b. 3:3:3:3
c. 4:5:4:5
40. In a golden rectangle, the
ratio of the length of the
rectangle to its width is
approximately 1.618:1.
Suppose a golden rectangle
has a length of 12
centimeters. What is its width
to the nearest tenth?
Pg 286 YOU SOLVE
38. The ratios of the
lengths of the sides of
three polygons are given
below. Make a conjecture
about identifying each
type of polygon.
a. 2:2:3
b. 3:3:3:3
c. 4:5:4:5
40. In a golden rectangle, the
ratio of the length of the
rectangle to its width is
approximately 1.618:1.
Suppose a golden rectangle
has a length of 12
centimeters. What is its width
to the nearest tenth?
Figures that are similar (~) have the same shape but not necessarily the same size.
Two polygons are similar
polygons if and only if
their corresponding
angles are congruent and
their corresponding side
lengths are proportional.
Example 1: Describing Similar Polygons
Identify the pairs of congruent angles and
corresponding sides.
#1
N  Q and P  R.
By the Third Angles Theorem, M  T.
Therefore: Tri NPM ~ Tri QRT
0.5
Identify the pairs of congruent angles and
corresponding sides.
#2
B  G and C  H.
By the Third Angles Theorem, A  J.
A similarity ratio (scale factor) is the ratio of the lengths of
the corresponding sides of two similar polygons.
The similarity ratio of ∆ABC to ∆DEF is
, or
The similarity ratio of ∆DEF to ∆ABC is
, or 2.
.
Example 2A: Identifying Similar Polygons
Determine whether the polygons are similar. If so, write the
similarity ratio and a similarity statement.
rectangles ABCD and EFGH
Example 2A Continued
Step 1 Identify pairs of congruent angles.
All s of a rect. are rt. s and are .
A  E, B  F,
C  G, and D  H.
Step 2 Compare corresponding sides.
Thus the similarity ratio is
,
and rect. ABCD ~ rect. EFGH.
Example 2B: Identifying Similar Polygons
Determine whether the
polygons are similar. If so,
write the similarity ratio and a
similarity statement.
Example 2B Continued
Step 1 Identify pairs of congruent angles.
P  R and S  W
isos. ∆
Step 2 Compare corresponding angles.
mW = mS = 62°
mT = 180° – 2(62°) = 56°
Since no pairs of angles are congruent, the triangles are not similar.
Check It Out! Example 2
Determine if ∆JLM ~ ∆SPN. If so, write the
similarity ratio and a similarity statement.
Step 1 Identify pairs of congruent angles.
N  M, L  P, S  J
Check It Out! Example 2 Continued
Step 2 Compare corresponding sides.
Thus the similarity ratio is
, and ∆LMJ ~ ∆PNS.
Example 3: Hobby Application
Find the length of the model to the nearest tenth of
a centimeter.
Let x be the length of the model in centimeters. The
rectangular model of the racing car is similar to the
rectangular racing car, so the corresponding lengths
are proportional.
Example 3 Continued
5(6.3) = x(1.8)
Cross Products Prop.
31.5 = 1.8x
Simplify.
17.5 = x
Divide both sides by 1.8.
The length of the model is 17.5 centimeters.
Check It Out! Example 3
A boxcar has the dimensions shown.
A model of the boxcar is 1.25 in. wide. Find the length of the model to the nearest
inch.
Check It Out! Example 3 Continued
1.25(36.25) = x(9)
Cross Products Prop.
45.3 = 9x
5x
Simplify.
Divide both sides by 9.
The length of the model is approximately 5 inches.
CW pg 285 prob 6 – 11 all; prob #12- 34 even
Due at end of class
Lesson Quiz: Part I
1. Determine whether the polygons are similar. If so, write the
similarity ratio and a similarity statement.
no
2. The ratio of a model sailboat’s dimensions to the actual boat’s
dimensions is
.
If the length of the model is 10 inches, what is the length of the actual
sailboat in feet?
25 ft
Lesson Quiz: Part II
3. Tell whether the following statement is sometimes, always, or never
true. Two equilateral triangles are similar.
Always