Geometry: Similarity NOTES
Proportions
A ratio is one thing compared to or related to another thing. It is a comparison of two
a
quantities. The ratio of a to b can be expressed as (so long as b ¹ 0), or a:b. The ratio of two
b
corresponding quantities is called a scaled factor.
A proportion is two ratios that have been set equal to each other; a proportion is an equation
that can be solved. In other words, an equation stating that ratios are equal is a proportion.
Every proportion has two cross products. The cross products mean multiply the denominator of
the second fraction and the numerator of the first fraction, AND multiply the denominator of
a c
the first fraction and the numerator of the second fraction. EX. = ® ad = bc
b d
These ratios are also proportional relationships. The two quantities vary directly with one and
other. If one item is doubled, the other, related item is doubled this is called a direct variation.
The cross multiplication creates extremes and means. The Extremes = ad, while the Means = bc.
This translates to: the product of the extremes = the product of the means.
When I say that a proportion is two ratios that are equal to each other, this mean two fractions
being equal to each other. For instance, 5/10 equals 1/2. Solving a proportion means that one
part of one of the fractions is missing, and you need to solve for that missing value.
EX.
x
50
= 12 ®2x = 50® x = 25.
Notice: the 2 is multiplied by the x, and the 50 is
multiplied by the 1 to create an equation without fractions. Dividing each side by 2
solves the remaining equation.
EX.
6
18.2
(
)
= 9y ®6 y = 9 18.2 ®6 y = 163.8 ® y = 27.3 Notice: the 6 is multiplied
by the y, and the 18.2 is multiplied by the 9 to create an equation without fractions.
Dividing each side by 6 solves the remaining equation.
EX.
4 x-5
3
(
) ( )
= -29
® 6 4x -5 = 3 -29 ® 24x - 30 = -87
6
® 24x = -57 ® x = -57
® x = - 198 = -2.375
24
Notice: the 6 is multiplied by the
(4x – 5), and the 3 is multiplied by the -29 to create an equation without fractions.
Solving the two-step equation, x = -2.375
EX. Monique randomly surveyed 30 students from her class and found that 18 had a dog or a
cat for a pet. If there are 870 students in Monique’s school, predict the total number of
18
x
students with a dog or a cat.
=
® x = 522
30 870
Geometry: Similarity NOTES
Properties of Similar Polygons
Two polygons are similar if their corresponding angles are congruent and the corresponding
sides have a constant ratio (they are proportional). Problems with similar polygons ask for
missing sides or lengths. To solve for a missing length, find two corresponding sides whose
lengths are known and write a proportional equation and solve for the variable.
Two figures that have the same shape are said to be similar. When two figures are similar, the
ratios of the lengths of their corresponding sides are equal. The symbol for similar is
.
∼
Triangles and polygons are similar IF AND ONLY IF their corresponding angles are congruent and
the measures of their corresponding sides are proportional. The ORDER of the vertices is
important because this identifies the corresponding angles and sides.
Triangles have specific similarity rules:
Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles
of another angle, then the triangles are similar
Side-Side-Side Similarity Theorem
If the measures of the corresponding sides of two
triangles are proportional, then the triangles are similar
Side-Angle-Side Similarity Theorem
If the measures of two sides of a triangle are
proportional to the measures of two corresponding sides
of another triangle and the included angles are
congruent, then the triangles are similar
Similarity involves PROPORTIONS.
If ΔABC ∼ ΔRST, list all pairs of congruent angles and write a proportion that relates the
corresponding sides. Congruent angles: ∠A ≅ ∠R, ∠B ≅ ∠S, ∠C ≅ ∠T;
Proportion:
AB
RS
,
BC
ST
,
AC
RT
***NOTE***: the parts of ONE triangle serve as the Numerator and the parts of ANOTHER serve as the denominator.
The concept of proportions you learned in elementary and middle school involves RATIOS.
EX. The number of students that participate in sports programs at Central High School is 520.
The total number of students in the school is 1850. Find the athlete-to-student ratio to the
nearest tenth.
Student athletes
Total students
520
52
= 1850
= 185
» 0.281 » 0.3
Geometry: Similarity NOTES
EX. The two polygons are similar. Find x & y.
Step 1: Write proportions for the sides of the polygon:
RS
AB
= TU
= VU
® 46 = 3x =
CD
ED
y+1
8
Step 2: Solve ONE proportion for the sides of the polygon:
RS
AB
= TU
® 46 = 3x ® x =
CD
9
2
Step 3: Solve the OTHER proportion for the sides:
RS
AB
= VU
® 46 =
ED
y+1
8
® y = 133
***NOTE***: the parts of ONE pentagon serve as the Numerator and the parts of ANOTHER serve as the denominator.
EX. Than is designing a new menu for the restaurant where he works. Determine whether the
following sizes for the new menu are similar to the original menu. If so, write the similarity
statement and scale factor. Explain your reasoning.
Original MENU
Choice A
Choice B
Geometry: Similarity NOTES
EX. If ABCDE ∼ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each
polygon.
Step 1: Write proportions for the sides of the polygon:
AE
VR
ED
DC
AB
DC
= VU
= AB
= UT
= BC
® 47 = 47 = 10.5
= 10.5
= ST6
RS
ST
Step 2: Solve ONE proportion for the sides of the polygon:
ED
VU
AB
= AB
® 47 = 10.5
® AB = 6
RS
AE
VR
ED
DC
6
DC
= VU
= AB
= UT
= BC
® 47 = 47 = 10.5
= 10.5
= ST6
RS
ST
Step 3: Solve the OTHER proportion for the sides:
AE
VR
ED
DC
6
DC
= VU
= AB
= UT
= BC
® 47 = 47 = 10.5
= 10.5
= ST6 ® DC = 6, ST = 10.5
RS
ST
Step 4: Substitute and solve for perimeters.
Pentagon ABCDE = 26
Pentagon RSTUV = 45.5
EX. Determine whether the triangles are similar. If so, write the similarity statement. Explain
your reasoning.
To determine similarity, calculate the missing angle
measures. ÐA = 180- ÐB - ÐC = 180- 42-58 .
ÐA = 80
Now, the triangles are similar, meaning:
ÐA @ ÐE, ÐB @ ÐD, ÐC @ ÐF .
DABC ∼ DDEF by AA Similarity.
EX. Determine whether the triangles are similar. If so, write the similarity statement. Explain
your reasoning.
To determine similarity, notice the parallel lines and
vertical angles.
.
Because the lines are parallel, alternate interior
angles are congruent: ÐQ @ ÐN .
DQPX ∼ DNMX by AA Similarity.
Geometry: Similarity NOTES
EX. Determine whether the triangles are similar. If so, write the similarity statement. Explain
your reasoning.
To determine similarity, notice vertical angles &
determine if the sides are proportional.
AC
AB .
ÐACB @ ÐDCE, DC
= BC
= DE
EC
Check the proportionality:
AC
DC
3
2
AB
= BC
= DE
® 64 = 7.5
= 69
EC
5
= 32 = 32
.
DACB ∼ DDCE by SSS Similarity.
EX. Determine whether the triangles are similar. If so, write the similarity statement. Explain
your reasoning.
To determine similarity, determine if the sides are
proportional and notice any shared corresponding
parts. ÐRMS @ NMP, MN
.
= MP
= NP
RM
SM
RS
Check the proportionality:
MN
RM
= MP
® 10
= 12
® 52 = 52 .
SM
25
30
DRMS ∼ DNMP by SAS Similarity.
EX. In ΔEFG, the ratio of the measures of the angles is 5:12:13, and the perimeter is 90
centimeters. Find the measures of the angles.
To determine the denominator, add the ratios: 5 + 12 + 13 = 30, now REWRITE the ratios as
fractions: 30 : 30 : 30 . These fractions represent the percentage of the Triangle’s sum
Since E is first, it corresponds with 5, F corresponds with 12, & 13 corresponds with G.
5
12 13
Set up your equations using your percentages and
,
5
30
(180) , (180) , : (180).
12
30
13
30
EX. STANDARDIZED TEST PRACTICE
If ΔRST and ΔXYZ are two triangles such that XY = 3 which of the following would be
sufficient to prove that the triangles are similar?
RS
RT
A. XZ
ST
= YZ
RS
B. XY
ST
= RT
= YZ
XZ
C.
2
ÐR @ ÐS
RS
D. ST
= XY
XZ
.
Geometry: Similarity NOTES
EX. SHORT RESPONSE TEST PRACTICE
Given RS || UT, RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10. Find RQ and QT. 8; 20
Notice the lines UT and RS are parallel. This means
ÐU @ ÐS, ÐT @ ÐR and ÐUQT @ ÐRQS because they are vertical
RS
angles. Set up the proportions: UT
(
RQ
x+3
= QT
® 104 = 2x+10
, now solve.
)
(
)
4 2x +10 = 10 x +3
8x + 40 = 10x + 30
2x = 10
x =5
Substitute 5 for x, RQ = 8 and QT = 20.
EX. Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole
and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured
the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height
of the Sears Tower?
2
STEP 1: write the proportions for the given triangles: 12
2x = 12 242
( )
STEP 2: Solve the proportion: 2x = 2904
x = 1452 ft
= 242
x