8.1Inclass
January 12, 2015
Geometry A Chapter 8
8.1 Ratio and Proportion
What is a ratio? What is a proportion?
Give an example of two ratios that reduce
to the same value
How do you solve a proportion?
ex: 3x + 2 = 5x - 1
4
6
In a proportion, the first and fourth terms are called
the extremes and the second and third terms are
called the means
Theorem 59: Means-Extremes Products
Theorem: the product of the means is equal
to the product of the extremes.
THEOREM 60: If the product of a pair of nonzero numbers is
equal to the product of another pair of nonzero numbers, then either
pair of numbers may be made the extremes, and the other pair the
means of a proportion. (MEANS-EXTREMES RATIO THEOREM
Find the ratio of x: y if
5x-6by=2cx+ya
8.1Inclass
January 12, 2015
Definition: If the means in a proportion are equal, then either mean
is called a geometric mean or mean proportional between the
extremes.
8/12=12/18
1/5=5/25
Can you find an example of a geometric means
proportion?
Find the geometric and arithmetic means between
6 and 36
8.1Inclass
January 12, 2015
8.2 Similar Polygons
def: Similar polygons are polygons in which
1) the ratios of the corresponding sides are equal
and
2) the measures of the corresponding angles are
congruent.
Can you create a figure that is three
times as
large as the given figure ?
8.1Inclass
January 12, 2015
Two figures are similar if one is a
reduction or a dilation of the other
D
(0, 6)
C
A
(0, 0)
B(8,0)
E
ΔAED is a dilation of
ΔABC in the ratio of
2:3. Find the
lengths of sides of
ΔAED (recall that in
a right Δ, a 2+b2=c2
8.1Inclass
January 12, 2015
Let these triangles
be similar with a
ratio of 3 to 5
x
2
z
5
4
y
Find the length of the missing
sides x, y , z AND find the
perimeters of both triangles and
their ratio
If two polygons are similar, then the ratio of their
sides is equal to the ratio of their perimeters.
12
10
y
8
4
x
what is the ratio of perimeters? Can you find the
perimeter of the second triangle without first determining
the lengths of the sides?
8.1Inclass
January 12, 2015
Solve for x and y, given that the triangles are similar
8.1Inclass
January 12, 2015
8.1Inclass
January 12, 2015
8.1Inclass
January 12, 2015
8.1Inclass
January 12, 2015
Assume that you are 5 ft. 4 in. tall and that you are standing 10 ft from the mirror.
Assume that the mirror is placed 18 ft from the foot of the tree.. Set up a proportion
to determine the height of the tree.
8.1Inclass
January 12, 2015
Congruences and proportions in similar triangles
8.4
If two triangles are congruent then their corresponding parts are
congruent- CPCTC
IF two triangles are similar then their corresponding parts are
similar? NO
IF two triangles are similar then (1) the corresponding sides are
proportional (ie. THE RATIOS OF THE MEASURES OF
CORRESPONDING SIDES ARE EQUAL) CSSTP
(2) the corresponding angles are congruent CASTC
8.1Inclass
January 12, 2015
E
B
A
C
GIVEN: <A = <D, <B = <E
PROVE: AB* EF = BC * DE
D
F
8.1Inclass
January 12, 2015
G
8
L
9
J
15
K
M
12
R
18
Q
P
GJKL ~ MQPR
find QP. PR, MR
8.5 Three Theorems Involving Proportions
Th 65: If a line is parallel to one side of a triangle and intersects the
other two sides, it divides those two sides proportionally.
called the SIDE-SPLITTER THM
G: BE // CD
P: AB/BC = AE/ED
8.1Inclass
January 12, 2015
TH 66: If three or more parallel lines are intersected by two
transversals, the parallel lines divide the transversals proportionally.
V
A
B
G: AV // BW // CY
W
C
Y
P: AB/BC = VW/WY
Can you see how to prove this using
side-splitter?
Th 67: If a ray bisects an angle of a triangle, it divides the opposite side
into segments that are proportional to the adjacent sides. (Angle Bisector
Theorem)
A
Given: AD bisects <CAD
1 2
B
Prove: AC/CD =AB/DB
D
C
<1 = <2
1
2
8.1Inclass
January 12, 2015
3
G: <3 = <5
P: RV = RS
VT ST
V
5 4
T
R
S
Find the perimeter of the
largest triangle
x-2
4
5
x+3
9
y
8.1Inclass
January 12, 2015
G: ABDF is a parallelogram
P:
CBD ~
DFE
B
A
F
E
D
C