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8.1 Ratio and Proportion
Ratio of a to b: the quotient a/b if a and b are two quantities that are measured in
the same units. It can also be written as a:b.
Ratios are generally expressed in simplified form. 6:8 is simplified to 3:4
Simplify the ratios.
60 cm
1m
8 in
1 ft
2 lb
8 oz
2640 ft
1 mile
1 kg
800 g
Using ratios
The perimeter of the isosceles triangle is 56 in. The ratio of LM:MN is 5:4. Find
the measure of each side of the triangle.
L
M
N
The measures of the angles in a triangle are in the ratio of 3:4:8. Find the
measures of the angles.
Proportion: an equation that equates two ratios.
Cross Product Property: the product of the means = the product of the extremes.
If a = c , then ad = bc
a and d are the extremes
b d
b and c are the means
Reciprocal Property: if two ratios are equal, then their reciprocals are also equal.
If a = c , then b = d
b d
a c
Solving Proportions
4=x
2 = 13
12 9
3 m
x+2=x
3
4
8 = 14
t 5
m – 5 = 25
4
10
The ratio of gravity of Venus to Earth is 9:10. If your math teacher weighs 200
pounds on Earth, how much would he weigh on Venus?
461: 1 – 19, 21 – 22, 25 – 50
8.2 Problem Solving in Geometry with Proportions
Additional Properties: if a = c, then a = b
b d
c d
if a = c, then a + b = c + d
b d
b
c
Using Properties of Proportions
In the diagram AB = AC. Find the length of BD.
BD CE
A
D
B
C
E
Geometric Mean: for two positive numbers a and b, the positive number x such
that a = x or x = ab.
x b
Find the geometric mean of the two numbers.
8 and 18
4 and 25
6 and 8
9 and 36
Using Proportions in Real Life
A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic
itself was 882.75 feet long. How wide was it?
Length of model = Width of model
Length of Titanic Width of Titanic
Or
Length of model = Length of Titanic
Width of model
Width of Titanic
468: 1 – 20, 23 – 28, 31 – 34
8.3 Similar Polygons
Similar Polygons: two polygons such that their corresponding angles are congruent
and the lengths of corresponding sides are proportional. The symbol is ~.
A
B
Q
R
ABCD ~ QRST
D
C
T
S
List all the pairs of congruent angles and write the ratios of the corresponding
sides: <A = < , <B = < , <C = < , <D = < , AB = BC = CD = DA
Decide whether the figures are similar. If so, write a similarity statement.
A
15
H
9
6
10
B
C
J
8
K
12
8
You have a picture that measures 3.5 inches by 5 inches and you want to enlarge it
to be 16 inches wide. How long should it be?
Theorem: If two polygons are similar, then the ratio of their perimeters is equal
to the ratios of their corresponding side lengths.
If KLMN ~ PQRS,
then KL + LM + MN + NK = KL = LM = MN = NK
PQ + QR + RS + SP PQ QR RS SP
Day One: 475: 1 – 7, 8, 10, 11 – 38 all
8.4 Similar Triangles
In the diagram, GED ~ GFH.
Write the statement of proportionality.
Find m<GFH and m<GHF
Find GF and FE
AA Similarity Postulate: if two angles of one triangle are congruent to two angles
of another triangle, then the two triangles are similar.
If <A = <D and <B = <E, then
ABC ~
DEF
Examples:
In the diagram LMN ~ PQN
m<M = ____, m<P = ____
MN = ____, QM = ____
Write a statement of proportionality: ------- = ------ = -----Example 4: page 482:
Use the proportion: f = n, to solve if g = 50, f = 8, and n = 3.
h g
Use the same proportion and solve if f = 10, n = 6, and g = 100.
Day One: 483: 1 – 8, 10 – 28 all
8.5 Proving Triangles are Similar
In the previous section we learned we can prove triangles are similar using the
Angle-Angle Similarity Postulate (AA). Today we find two more ways to prove
triangles are similar.
SSS: Side-Side-Side Theorem: If the lengths of the corresponding sides of two
triangles are proportional, then the triangles are similar.
If AB = BC = CA
DE EF FD
then
ABC ~
DEF
SAS: Side-Angle-Side Theorem: If two sides of one triangle are proportional to
two sides of another triangle and the included angles are congruent, then the two
triangles are similar.
If
AB = BC and m<B = m<E, then ABC ~
DE EF
DEF
Which of the following triangles are similar?
To determine the answer, compare short, middle, and long sides.
Repeat the previous example with triangles with the following measures:
Triangle One: 6, 15, 18; Triangle Two: 3, 5, 7; Triangle Three: 10, 25, 35
In the figure AC = 6, AD = 10, BC = 9, and BE = 15. How can you prove the triangles
are similar?
A
E
C
B
D
Examples 5 and 6 on page 491:
492: 1 – 29 all: you need rulers and protractors
8.6 Proportions and Similar Triangles
Triangle Proportionality Theorem: if a line parallel to one side of a triangle
intersects the other two sides, then it divides the two sides proportionally.
If QS ll TU, then RT = RU
TQ US
Converse of Triangle Proportionality Theorem: if a line divides two sides of a
triangle proportionally, then it is parallel to the third side.
If RT = RU , then QS ll TU
TQ US
If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
If r ll s and s ll t and l and m intersect
r, s, and t, then UW = VX.
WY XZ
If a ray bisects an angle of a triangle,
then it divides the opposite sides into
segments whose lengths are proportional
to the lengths of the other sides.
If CD bisects <ACB, then AD = CA
DB CB
Examples 1, 2, 3, and 4 on pages 498 – 500.
502: 1 – 30, 36, 37, 39 all