U SING S IMILARITY T HEOREMS THEOREM S THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the...
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Transcript of U SING S IMILARITY T HEOREMS THEOREM S THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem If the...
USING SIMILARITY THEOREMS
THEOREM S
THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar.
If = =A BPQ
BCQR
CARP
then ABC ~ PQR.
A
B C
P
Q R
USING SIMILARITY THEOREMS
THEOREM S
THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.
then XYZ ~ MNP.
ZXPM
XYMN
If X M and =
X
Z Y
M
P N
Proof of Theorem 8.2
GIVEN
PROVE
= = STMN
RSLM
TRNL
RST ~ LMN
SOLUTION
Paragraph Proof
M
NL
R T
S
P Q
Locate P on RS so that PS = LM.
Draw PQ so that PQ RT.
Then RST ~ PSQ, by the AA Similarity Postulate, and .= = ST SQ
RS PS
TR QP
Use the definition of congruent triangles and the AA Similarity Postulate to conclude that RST ~ LMN.
Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem,
it follows that PSQ LMN.
E
F D8
6 4A C
B
12
6 9
G J
H
14
6 10
Using the SSS Similarity Theorem
Which of the following three triangles are similar?
SOLUTION To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios of Side Lengths of ABC and DEF
= = , 6 4
AB DE
3 2
Shortest sides
= = , 12 8
CA FD
3 2
Longest sides
= = 9 6
BC EF
3 2
Remaining sides
Because all of the ratios are equal, ABC ~ DEF
= = ,1214
CA JG
67
Longest sides
E
F D8
6 4
Using the SSS Similarity Theorem
A C
B
G J
H
Which of the following three triangles are similar?
SOLUTION To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios of Side Lengths of ABC and GHJ
12 14
6 6 109
= = 1, 6 6
AB GH
Shortest sides
= 910
BC HJ
Remaining sides
Because all of the ratios are not equal, ABC and DEF are not similar.
E
F D8
6 4A C
B
12
6 9
G J
H
14
6 10
Since ABC is similar to DEF and ABC is not similar to GHJ, DEF is not similar to GHJ.
Using the SAS Similarity Theorem
Use the given lengths to prove that RST ~ PSQ.
SOLUTION PROVE RST ~ PSQ
GIVEN SP = 4, PR = 12, SQ = 5, QT = 15
Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides.
= = = = 4 SR SP
16 4
SP + PR SP
4 + 12 4
= = = = 4 ST SQ
20 5
SQ + QT SQ
5 + 15 5
Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that RST ~ PSQ.
The side lengths SR and ST are proportional to the corresponding side lengths of PSQ.
12
4 5
15
P Q
S
R T
USING SIMILAR TRIANGLES IN REAL LIFE
SCALE DRAWING As you move the tracing pin of a pantograph along a figure, the pencil attached to the far end draws an enlargement.
Using a Pantograph
P
R
T
S
Q
USING SIMILAR TRIANGLES IN REAL LIFE
Using a Pantograph
As the pantograph expands and contracts, the three brads and the tracing pin always form the vertices of a parallelogram.
P
R
T
S
Q
USING SIMILAR TRIANGLES IN REAL LIFE
Using a Pantograph
The ratio of PR to PT is always equal to the ratio of PQ to PS. Also, the suction cup, the tracing pin, and the pencil remain collinear.
P
R
T
S
Q
You know that . Because P P, you can apply the
SAS Similarity Theorem to conclude that PRQ ~ PTS.
= PQPS
PRPT
Using a Pantograph
S
R
Q T
P
How can you show that PRQ ~ PTS?
SOLUTION
SOLUTION Because the triangles are similar, you can set up a proportion to find the length of the cat in the enlarged drawing.
Using a Pantograph
10"
2.4"10"
S
R
Q T
P
= RQTS
PRPT
Write proportion.
=1020
2.4TS
=TS 4.8
Substitute.
Solve for TS.
So, the length of the cat in the enlarged drawing is 4.8 inches.
In the diagram, PR is 10 inches and RT is 10 inches. The length of the cat, RQ, in the original print is 2.4 inches. Find the length TS in the enlargement.
Finding Distance Indirectly
Similar triangles can be used to find distances that are difficult to measure directly.
ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.
85 ft6.5 ft
5 ft
A
B
C E
DUse similar triangles to estimate the height of the wall.
Not drawn to scale
Finding Distance Indirectly
85 ft6.5 ft
5 ft
A
B
C E
D
Use similar triangles to estimate the height of the wall.
SOLUTION
Using the fact that ABC and EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.
Due to the reflective property of mirrors, you can reason that ACB ECD.
85 ft6.5 ft
5 ft
A
B
C E
D
DE65.38
Finding Distance Indirectly
Use similar triangles to estimate the height of the wall.
SOLUTION
= ECAC
DEBA
Ratios of lengths of corresponding sides are equal.
Substitute.
Multiply each side by 5 and simplify.
DE5
= 856.5
So, the height of the wall is about 65 feet.