Chapter 8
8.1 Ratio and Proportional
8.2 Similarity
8.3 Methods of Proving Triangles Similar
8.4 Congruence and Proportions in Similar Triangles
8.5 Three theorems involving Proportions
8.1 Ratio and Proportion
This section reviews your previous knowledge of proportions.
Definitions
Ratio – is a comparison of two numbers
A ratio may be written as a fraction or with a colon
Equivalent ratios are two ratios that reduce to the same number
Ex: 9/12 and 6/8 are equivalent ratios because they both reduce to 3/4
A Proportion is an equation relating two equivalent ratios
Ex: 
or
this may be written as 9 : 12 = 3 : 4
5 Methods of manipulating a proportion
1.
If 
,
then ad = bc . a and d are referred to as the extremes and b
and c are referred to
as the means.
This is referred to as the Means- Extremes Product theorem which you probably learned in Pre-Alg as cross multiplication. It is used mainly to solve for a missing variable in a proportion.
Ex: 
9(4x)=12(x+4)
--- 36x=12x+48 --- 24x = 48 --- x = 2
2. 
You can flip a proportion and it remains true.
ex: If 3/4 = 1/x then 4/3=x Both sides of the proportion can be flipped and it is still a true equation
3.
You can compare the original proportions sideways and it is still a true
proportion.
ex. If 1/2 = 3/6 then 1/3 = 2/6
4. You can add the denominator to the numerator of both sides and the proportion is still true. That is because it is equivalent to adding one to both sides of the equations.
5. You can add the numerators and denominators together and you will get a fraction equivalent to an original.
Arithmetic and Geometric Means
Arithmetic means are just averages. Given two numbers a and b, to find the arithmetic mean
Geometric Means - if a number occurs twice in a proportion as the denominator of the first ratio and the numerator of the second ratio, it is referred to as a geometric mean
To find the geometric mean between two numbers take the square root of the product.
** notice that the principal or positive root is taken although in algebra it could be either. Since we will be dealing in measurement the only one we are interested in is the positive
Note also that you will have to simplify some radicals. One of the rules for radicals is that no factor of the radicand can be a perfect square.
Find the geometric mean between 12 and 6.
8.2 Similar Figures
Similar polygons have corresponding congruent angles and corresponding proportional sides.
The symbol for similar is ~ .
If two figures are similar, they are named so that the corresponding congruent angles and proportional sides line up.
The ratio between corresponding sides of similar figures are equivalent.
This ratio is referred to as the scale factor between the figures.
Missing sides can be found by setting up a proportion with the scale factor and solving.
Find the scale factor.
Solve for x and y.
When a figure is transformed using a dilation, the new figure is similar as it has the same angles, but the sides are proportionally larger.
When a figure is transformed by a reduction, the new figure is similar, but the sides are proportionally smaller.
A pinhole camera produces a reduced image of a candle that is 3 cm. The distance from the candle to the pinhole is 15 cm and the distance from the pinhole to the image is 5 cm. What is the height of the candle.
Sometimes a dilation is made on a coordinate system. Most times
it is made in relation to the origin and one similar figure overlaps the other.
If ABCD ~ FEHG, find the perimeter of ABCD.
You can find each individual side, but there is a theorem that states as follows.
The ratio of the perimeters of two similar polygons equals the ratio of any pair of corresponding sides.
This is much easier than dealing with all of the fractions.
8.3 Methods of Proving Triangles Similar
There are three methods you will use to prove that triangles are similar.
AAA~ Postulate - If three angles of one triangle are congruent to three angles of a 2nd triangle then the triangles are similar.
Example:
:
By the third angle theorem if you knew two pair of corresponding angles were congruent you could easily state the third angles are congruent – so you will rarely use AAA~ Postulate and instead use AA~ Theorem
AA~ states that if 2 angles of one triangle are congruent to two angles of a 2nd triangle then the triangles are similar.
SAS ~ If two sides of one triangle are proportional to the corresponding two sides of a 2nd triangle and their included angles are congruent, then the triangles are similar.
The vertical angles are congruent, the sides are proportional 4/6 = 8/12
Triangle ABE ~ Triangle DCE
SSS~ If 3 sides of one triangle are proportional to three sides of a 2nd triangle then the two triangles are similar.
8.4
Congruence and Proportions in Similar Triangles
In this section we have to take the proofs past just
proving the triangle similar and set up proportions between the sides or
manipulate the proportions using the 5 methods.
Once you have proven triangles similar you can state the proportions between the 3 sets of corresponding sides and the reason for the sides being proportional is CSSTP - Corresponding sides of similar triangles are proportional. (Def of ~ Triangles)
Example:
The second type of problem
in this section is the shadow problem which is based on similar triangles.
Given the shadows of two figures and the height of one of
the figures at the same time of day you can set up proportional sides between
the heights to figure out the missing length.
Example: John is 6 ft tall and casts a shadow of 10 feet at the same time that a flag pole casts a shadow of 50 feet. How tall is the flag pole?
The second type of shadow problem is when there is a street
light and we know the distance
a person is standing from the street light.
Ex: John is 6 feet tall and is standing 40 feet from a street light and
casting a shadow of 8 feet. How tall
is the street light?
Note: Be careful to add the distance that John is standing from the street
lamp to his shadow length so you have the whole
base of the larger triangle.
8.5 3 Proportion Theorems
Side Splitter - If a line is drawn parallel to one side of a triangle and
intersects the other two sides, it divides the two sides proportionally.
Ex. 1
Ex. 2
Ex. 3
The one thing you must be careful of is if the parallel
sides are involved you cannot use the side splitter, but must use the similar
triangles involved
Parallels Proportion Theorem
If 3 or more parallel lines are drawn, they divide each transversal
proportionally.
Ex. 1
Ex. 2
Angle Bisector Proportionality Theorem.- The bisector of an angle of a
triangle divides the opposite side into segments proportional to the two sides
of the angle.
Ex. 1
Ex. 2
A.M.=16
1. Find the geometric mean
and arithmetic mean between
18 2. 12 is the geometric mean between 8 and what number.
3.
DTUV
~ D
_URV__~
D___TRU__
|
___
4. If
, find the ratio of x to
y.
5.
If
, then
?
____32________6. A train has a length of 8 yards. If a model of the train is made with a ratio of 1 inch: 9 inches. Find the length of the model
|
______36____8. A triangle has sides of 9, 11,12 and a second triangle has a perimeter of 128. If the two triangles are similar, what is the length of the shortest side of the second triangle?
For each pair of triangle indicate if the triangles are similar and if so the reason why.
 |
SAS~
9.
SSS~ 11.
.
| SAS~ 12 |  |
|||
| NO 13. |
|
Solve for the indicated variable.
x =
12
14. 
x= 24 15.
Statements Reasons
XS and RY are
altitudes Given
XS ┴TR , RY ┴TS
Def of alt.
are
right angles Def of Perpendicular
All
right angles are congruent
Reflexive
Property
AA~
CSSTP
or Def of ~Triangles
