Similar Experiences
Similar Game Plans
Similar Characters
Similarity in Geometric Figures
Similarity in Geometry:
- The concept that shapes or other relations exist in specific ratios or
proportions to one another.
Ratios: When two quantities are measured in the same units, (or can be
converted to the same unit of measure), they can be expressed in a
constant relation to one another, referred to as a ratio.
Ratios are written: a / b or a:b
Simplifying Ratios:
Many times, the unit of measure provided will not always be the same. In
those cases they need to be converted in order to specify a meaningful
ratio.
12cm
12 cm
4m = 4* 100cm
12
400
3
100
6ft
18 in.
6 * 12 in.
= 18 in.
How might we convert these to simplified / meaningful ratios?
72
18
4
1
Using Ratios To Solve Problems
D
C
The perimeter of a rectangle ABCD is 60 centimeters.
The ratio of AB:BC is 3:2. Find the length and width of
the rectangle.
Hint: Because the ratio o AB:BC is 3:2, we can
represent the length of AB as 3x and BC as 2x
w
A
2 l + 2w = P
2(3x) + 2(2x) = 60
6x + 4x = 60
10x = 60
X=6
Therefore, ABCD has a length of 18 cm and a width of 12 cm
B
l
Using Ratios To Solve Problems
Extended Ratios: Ratios can extend beyond a simple
relationship of two measures to incorporate a third, or
more, provided they are all in a constant relation to one
another.
2x
In the triangle ABC, the angles exist in the
following extended ratio, 1:2:3.
How can we use the extended ratio to determine
the measure of the angles?
x + 2x + 3x = 180o
6x = 180
X = 30
Triangle Sum Theorem
Therefore the angle measures are 30o, 60o, and 90o
3x
x
Using Ratios To Solve Problems
The ratios of the side lengths of triangle DEF to the
corresponding side lengths of triangle ABC are 2:1.
How can we use this information to find the unknown
lengths?
C
3
A
DE is twice AB. DE = 8, so AB = ½ (8) = 4
B
F
Using the Pythagorean Theorem, we can
determine BC = 5
DF is twice AC. AC = 3, so DF = 6
EF is twice BC. BC = 5, so EF = 10.
D
8
E
Using Proportions
An equation that equates two ratios is a proportion.
For example, if the ratio a/b is equal to c/d, then the following proportion can be
written: a/b = c/d
The numbers “a” and “d” are referred to as the extremes.
The numbers “b” and “c” are referred to as the means.
Means
Extremes
a
b
c
= d
Properties of Proportions:
1. Cross Product Property – The product of the extremes equals the product of the
means.
If a/b = c/d, then ad = bc
2. Reciprocal Property – If two ratios are equal, then their reciprocals are also equal.
If a/b = c/d, then b/a = d/c
Solving Proportions
Using the Properties of Proportions, solve the following:
4/x=5/7
1. Using Cross Products:
5x = 28
X = 28/5
2. Using Reciprocal Property
4/x = 5/7
x/4 = 7/5
x = 4(7/5)
x = 28/5
3
=2
y+2
y
1. Using Cross Products
3y = 2(y + 2)
3y = 2y + 4
y=4
What do these all have in common?
Additional Properties of Geometry in Proportions
Properties of Proportions:
1. Cross Product Property – The product of the extremes equals the product of the
means.
If a/b = c/d, then ad = bc
2. Reciprocal Property – If two ratios are equal, then their reciprocals are also equal.
If a/b = c/d, then b/a = d/c
3. If a/b = c/d, then a/c = b/d
4. If a/b = c/d, then a+b/b = c+d/d
Problem Solving in Geometry with Proportions
A
In the diagram AB / BD = AC / CE.
Find the length of BD.
16
B
x
Given: AB _ AC
BD -- CE
C
10
D
AB _ AC
BD -- CE
E
16+x/x = 30/10
16/x = (30-10)/10
16/30-10 = x/10
16/x = 20/10
20x = 160
20x = 160
X=8
X=8
30
30x = 10(16+x)
30x = 160+10x
20x = 160
X=8
Geometric Mean
Geometric Mean:
The geometric mean of two numbers “a” and “b” is the positive number x such that:
a/x = x/b.
If you solve this proportion for x, through cross multiplication we find
that:
x2 = a*b, therefore
x = \/ a* b
Example:
What is the Geometric Mean of 8 and 18?
8/x = x/18  x2 = 144  x = 12
Also, 8/12 = 12/18 because
\/ 8 * 18 = \/ 144 = 12
Using Geometric Mean
International Paper Standards set
a standard ratio for length and
width for all writing paper which is
generally recognized around the
world. Two paper types A3 and A4
are shown to the right. The length
represented by “x” is the geometric
mean of 210mm and 420mm. Find
the value of x.
A4
x
A3
210 mm
x
210 / x = x / 420
X2 = 210 * 420
X = \/ 210 * 420
X = \/ 210 * 210 * 2
X = 210 \/ 2
420mm
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?
Width of ship = x
X ft / 11.25 in = 882.75 ft / 107.5 in
X = (11.25 * 882.75) / 107.5
X = 92.4 Feet
Proportions and Similar Triangles
R
Triangle Proportionality Theorem:
If a line parallel to one side of a triangle
intersects the other two sides, then it divides
the two sides proportionately
T
U
>
If TU || QS, then RT/TQ = RU/US
Q
Theorem 8.6
If thee parallel lines intersect two transversals,
then they divide the transversals proportionately.
If r|| s and s|| t, then l and m intersect r,
s, t and t, then UW/WY = VX = XZ
S
>
l
U
W
Y
m
X
Z
V
t
r
s
Theorem 8.7
If a ray bisects an angle of a triangle , then it
divides the opposite side into segments whose
lengths are proportional to the lengths of the
other two sides.
If CD bisects <ABC, then AD/DB = CA/CB
A
D
C
B
Using Proportions in Similar Triangles
C
In the diagram, AB||ED, BD = 8, DC = 4,
and AE = 12.
What is the length of EC?
4
E
D
12
8
A
B
G
21
M
56
Given the diagram, determine if MN || GH.
H
16
~ <3. What is
In the diagram, <1 ~= <2, <2 =
the length of TU?
N
48
P
S
9 1
Q
15 2
R
3
L
11
T
U