Chapter Seven Similar Polygons Ruby Weiner & Leigh Zilber.
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Transcript of Chapter Seven Similar Polygons Ruby Weiner & Leigh Zilber.
Chapter SevenSimilar Polygons
Ruby Weiner & Leigh Zilber
7.1 Ratios and Proportions
• Ratio: the quotient of 2 values A
B
C
D
60
30
90 90
60
30
1) Find the ratio of AE to BE10: 5x 2:x
2) Find the ratio of largest > of triACE to smallest > of triBDE90:30 3:1
E
10
5x
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?
4
workout 1) 3x + 2x = 75x = 7 --> 7/53(7/5) = 21/5 meters2(7/5) = 14/5 meters
2) x/325 = 32/100100x = 325 x 32100x = 10400 x = 104
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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 examples: (given a/b = 3/5)
1. 5a = 3b2. 5/b = 3/a3. a+b/b = 3+5/5 --> 8/54. 5/3 = b/a
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Proportion Practice Problems
• Choose yes or no– given: 10/20 = a/b
– 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?
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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
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7.3 Similar Polygons• recall congruent triangles
– corresponding angles --> congruent– corresponding sides --> congruent
Similar triangles
A
B
C D
E
F
-Corresponding angles are congruent -Corresponding sides are in proportion-AB/DE = BC/EF = AC/DF
9
Examples: find length of EF if triABC is similar to triDEF
A
B
C
D
E
F
2
3
4 4
6
x
2/4 = 4/x 2x = 16x = 8
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?
Conclude: yes, AA Similarity (AA~)
CA
B
Practice Problemso Determine if the triangles are similar and how.1)
2) Given: Both Triangles are Isosceles
50
40
555 5
Answers
• 1. 40 + 90 + x = 180 x = 5050 + 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.
Example: Are these Triangles congruent?Why?Answer: Yes, SAS~
1x y
E F
A
B C
Given: Angle A is congruent to Angle D AB/DE = AC/DF
7-5 Continued
• Theorem 7-2 (SSS Similarity Theorem)- If the sides of two triangles are in proportion, then the
triangles are similar. Example:
Answer: Yes, SSS~ A
BC
D
E F
1X Y
Given: AB/DE = BC/EF = AC/DF
Practice Problems
• 1.
• 2.
6
9
A
B
E
D
C
10
15
7.5
65K 12 M
LP
N
O
658
5
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 ofA) TN/ NRB) TR/NRC) RN/RT M N
R
S T
6
3Answer: a) tn/nr = sm/mr = 3/6 = ½b) Tr/nr = sr/mr = 9/6 = 3/2 c) Rn/rt = rm/rs = 6/9 = 2/3
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:
12
F
E
G
D
K
3
4Given: Triangle DEF; Ray DG bisects Angle FDEProve: GF/ GE = DF/DE
Answer: (Plan for Proof)Draw a line through E parallel to Ray DGAnd 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:
an
g
b
Answer
• 1. a/n = b/g
Practice Proof
• Given: Angle H and Angle F are right triangles Prove: HK * GO = FG * KO
Statements Reasons 1
2O
K
H
F G
Answer to Proof
Statements Reasons
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/GO7. HK*GO = FG*KO
1. Vertical Triangles are congruent.
2. Given3. Def. of right
triangle.4. Def. of congruent
triangle5. AA~6. Corr. Sides of ~
Triangles are in proportion.
7. A property of proportions.