Transcript Document 7185839

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.
P
A
Q
R
If AB = BC = CA
RP
PQ
QR
then ABC ~ PQR.
B
C
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.
If
X
XY
M and ZX = MN
PM
then XYZ ~ MNP.
M
X
P
Z
Y
N
Proof of Theorem 8.2
GIVEN
RS
= ST = TR
LM
MN
NL
PROVE
 RST ~  LMN
SOLUTION Paragraph Proof
M
L
S
P
N
Q
R
T
Locate P on RS so that PS = LM.
Draw PQ so that PQ
RT.
Then  RST ~  PSQ, by the AA Similarity Postulate, and RS = ST = TR .
PS
SQ
QP
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.
Use the definition of congruent triangles and the AA Similarity Postulate to
conclude that  RST ~  LMN.
Using the SSS Similarity Theorem
Which of the following three triangles are similar?
12
A
E
C
6
6
9
F
14
G
J
4
8
D
6
B
10
H
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
AB
6
3
=
=
,
DE
4
2
Shortest sides
CA
12
3
=
=
,
FD
8
2
Longest sides
BC
9
3
=
=
EF
6
2
Remaining sides
Because all of the ratios are equal,  ABC ~  DEF
Using the SSS Similarity Theorem
Which of the following three triangles are similar?
12
A
E
C
6
6
9
F
14
G
J
4
8
D
6
B
10
H
SOLUTION
To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios
Lengths
of  ABC
and and
GHJ ABC is not
SinceofSide
ABC
is similar
to  DEF
AB
6 to  GHJ,  DEF
CA is12
6
similar
not= similar
=
,
=
=
1
,
GH
6
JG
14
7
Shortest sides
Longest sides
BC .
to  GHJ
=
9
HJ
10
Remaining sides
Because all of the ratios are not equal,  ABC and  DEF are not similar.
Using the SAS Similarity Theorem
Use the given lengths to prove that  RST ~  PSQ.
SOLUTION
GIVEN
SP = 4, PR = 12, SQ = 5, QT = 15
PROVE
 RST ~  PSQ
S
Use the SAS Similarity Theorem.
Find the ratios of the lengths of the corresponding sides.
Paragraph Proof
4
P
SR
SP + PR
4 + 12
16
=
=
=
= 4
SP
SP
4
4
ST
SQ + QT
5 + 15
20
=
=
=
= 4
SQ
SQ
5
5
12
R
The side lengths SR and ST are proportional to the corresponding
side lengths of  PSQ.
Because S is the included angle in both triangles, use the
SAS Similarity Theorem to conclude that  RST ~  PSQ.
5
Q
15
T
USING SIMILAR TRIANGLES IN REAL LIFE
Using a Pantograph
SCALE DRAWING As you move the tracing pin of a pantograph along
a figure, the pencil attached to the far end draws an enlargement.
P
R
Q
T
S
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
Q
T
S
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
Q
T
S
Using a Pantograph
P
R
How can you show that  PRQ ~  PTS?
T
Q
SOLUTION
S
PR
PQ
You know that PT = PS . Because P  P, you can apply the
SAS Similarity Theorem to conclude that  PRQ ~  PTS.
Using a Pantograph
P
In the diagram, PR is 10 inches and
RT is 10 inches. The length of the cat,
RQ,
original
print
2.4 inches.
Find in
thethe
length
TS in
theisenlargement.
10"
R
2.4"
10"
T
Q
SOLUTION
Because the triangles are similar, you can set up a proportion
to find the length of the cat in the enlarged drawing.
PR RQ
=
Write proportion.
PT TS
10 = 2.4
Substitute.
20
TS
Solve for TS.
TS = 4.8
So, the length of the cat in the enlarged drawing is 4.8 inches.
S
Finding Distance Indirectly
ROCK CLIMBING
are at an
indoor
climbing
wall.
To estimate
of
SimilarYou
triangles
can
be used
to find
distances
thatthe
areheight
difficult
the wall, youto
place
a mirror
on the floor 85 feet from the base of the wall. Then
measure
directly.
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.
Use similar triangles to estimate
the height of the wall.
D
B
5 ft
Not drawn to scale
A
6.5 ft
C
85 ft
E
Finding Distance Indirectly
Use similar triangles to estimate
the height of the wall.
SOLUTION
Due to the reflective property of mirrors,
you can reason that ACB 
ECD.
D
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.
B
5 ft
A
6.5 ft
C
85 ft
E
Finding Distance Indirectly
Use similar triangles to estimate
the height of the wall.
SOLUTION
DE EC
Ratios of lengths of
=
corresponding sides are equal.
BA AC
So,
the height of the wall is about 65 feet.
DE = 85
Substitute.
5
6.5
D
Multiply each side by
5 and simplify.
65.38  DE
B
5 ft
A
6.5 ft
C
85 ft
E