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Chapter Seven
Similar Polygons
Ruby Weiner & Leigh Zilber
7.1 Ratios and Proportions
• Ratio: the quotient of 2 values
D
A
60
60
E
90 90
30
C
30
B
1) Find the ratio of AE to BE
10: 5x 2:x
1) Find the ratio of largest > of triACE
to smallest > of triBDE
90:30 3:1
Ratio Practice Problems
• A telephone pole 7 meters tall snaps into 2
parts. The ratio of the 2 parts is 3:2. Find the
length of each part.
• A teams best hitter has a life time batting
average of .320. He has been at bat 325 times.
– how many hits has he made?
workout
1) 3x + 2x = 7
5x = 7 --> 7/5
3(7/5) = 21/5 meters
2(7/5) = 14/5 meters
2) x/325 = 32/100
100x = 325 x 32
100x = 10400
x = 104
4
7.2 Properties of Proportions
• Proportion: equation stating that 2 ratios are equal
• Properties (given a/b = c/d) :
– b/a = d/c
– ad = bc
– a/c = b/d
– a+b/b = c+d/d
5
examples: (given a/b = 3/5)
1. 5a = 3b
2. 5/b = 3/a
3. a+b/b = 3+5/5 --> 8/5
4. 5/3 = b/a
Proportion Practice Problems
• Choose yes or no
– given: 10/20 = a/b
–
–
–
–
–
6
is 10 x b = 20 x a ?
is 10/20 = b/a ?
is 30/20 = a+b/b ?
is 20/10 = b/a ?
10/a = b/20?
ANSWERS
YES b.c ad = bc
NO b.c a/b no= d/c
YES b.c a+b/b = c+d/d
YES b.c b/a = d/c
NO b.c a/c no= d/b
7
7.3 Similar Polygons
•
recall congruent triangles
– corresponding angles --> congruent
– corresponding sides --> congruent
Similar triangles
E
B
A
8
C D
-Corresponding angles are congruent
-Corresponding sides are in proportion
-AB/DE = BC/EF = AC/DF
F
Examples: find length of EF if triABC is similar to triDEF
E
B
2
A
4
4
3
C
D
2/4 = 4/x
2x = 16
x=8
9
x
6
F
7-4 A Postulate for Similar Triangles
•
Postulate 15: AA Similarity Postulate
- If two angles of one triangle are congruent to two angles of
another triangle, then the triangles are similar.
Example: Are these triangle similar? How?
B
A
Conclude: yes, AA Similarity (AA~)
C
Practice Problems
o Determine if the triangles are similar and how.
1)
50
40
2) Given: Both Triangles are Isosceles
5
5
5
5
Answers
• 1. 40 + 90 + x = 180 x = 50
50 + 90 + x = 180 x = 40
they are similar by AA similarity
7-5 Theorems for Similar Triangles
• Theorem 7-1 (SAS Similarity Theorem)
- If an angle of one triangle is congruent to an angle of another
triangle and the sides including those angles are in
proportion, then the triangles are similar.
A
Example: Are these
Triangles congruent?
Why?
Answer: Yes, SAS~B
Given: Angle A is congruent to Angle D
AB/DE = AC/DF
x
C
E
1
y
F
7-5 Continued
• Theorem 7-2 (SSS Similarity Theorem)
- If the sides of two triangles are in proportion, then the
triangles are similar.
Example:
A
D
Answer: Yes, SSS~
X
Given:
AB/DE = BC/EF = AC/DF
B
C
E
1
Y
F
Practice Problems
E
• 1.
10
6
C
B
15
D
9
A
• 2.
L
8
O
7.5
5
N
65
K
P
65
12
M
Answers
• 1. Triangle BAC ~ Triangle EDC; SAS~
• 2. Triangle LKM ~ Triangle NPO; SAS~
7-6 Proportional Lengths
• Theorem 7-3 (Triangle Proportionality Theorem)
- If a line parallel to one side of a triangle intersects the other
two sides, then it divides those sides proportionally.
Example:
Find the numerical value of
R
A) TN/ NR
B) TR/NR
6
C) RN/RT
M
N
Answer:
a) tn/nr = sm/mr = 3/6 = ½
b) Tr/nr = sr/mr = 9/6 = 3/2
c) Rn/rt = rm/rs = 6/9 = 2/3
3
S
T
7-6 Continued
• Corollary
- If three parallel lines intersect two
transversals, then they divide the transversals
proportionally.
• Theorem 7-4 (Triangle Bisector Theorem)
- If a ray bisects an angle of a triangle, then it
divides the opposite side into segments
proportional to the other two sides.
7-6 Continued
Example:
F
G
2
D
Given: Triangle DEF; Ray DG
bisects Angle FDE
Prove: GF/ GE = DF/DE
1
3
4
K
E
Answer: (Plan for Proof)
Draw a line through E parallel to Ray DG
And intersecting Ray FD at K.
Apply Triangle Proportionality Theorem
To Triangle FKE. Triangle DEK is isosceles
With DK = DE. Substitute this into your
Proportion to complete the proof.
Practice Problems
1. State a proportion for the diagram:
n
a
g
b
Answer
• 1. a/n = b/g
Practice Proof
• Given: Angle H and Angle F are right triangles
K
Prove: HK * GO = FG * KO
Statements
Reasons
H
1
O
2
F
G
Answer to Proof
Statements
1. Angle 1 is congruent
to Angle 2.
2. Angle H and Angle F
are right Triangles.
3. Angle H = 90 and
Angle F = 90
4. Angle H is congruent
to Angle F.
5. Triangle HKO ~
Triangle FGO
6. HK/FG = KO/GO
7. HK*GO = FG*KO
Reasons
1. Vertical Triangles
are congruent.
2. Given
3. Def. of right
triangle.
4. Def. of
congruent
triangle
5. AA~
6. Corr. Sides of ~
Triangles are in
proportion.
7. A property of
proportions.