Transcript 8.5 day 2 - Bismarck Public Schools

Chapter 8
Similarity
Section 8.5
Proving Triangles are Similar
USING SIMILARITY THEOREMS
USING SIMILAR TRIANGLES IN REAL LIFE
USING SIMILARITY THEOREMS
Postulate
C
E
D
A D and C F
ABC ~ DEF
F
A
B
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
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 SIMILARITY THEOREMS
Determine if the triangles are similar
Compare Side Lengths of LKM and NOP
15
18 6
2
3
5
7 3
Ratios Different, triangles not similar
USING SIMILARITY THEOREMS
Determine if the triangles are similar
Compare Side Lengths of LKM and NOP
18 3 6 3 15 3



30 5 10 5 25 5
Ratios Same, triangles are similar
RQS ~ LKM
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
USING SIMILARITY THEOREMS
CED
44°
68°
20
5
2
USING SIMILARITY THEOREMS
Statements
Reasons
USING SIMILARITY THEOREMS
Statements
~
Reasons
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
Finding Distance Indirectly
6 2

2 x  144
x 24
The Tree is 72 feet tall
x  72
Finding Distance Indirectly
4 2
6 2
4 x  144
 x
72
2 x  144
x 24
The Tree is 72 feet tall
x  36
x  72
72
4
x
The mirror would need to be placed 36 feet from the tree
HW
Pg :6;9;11;13-17;19-25;27-29;3234;39-47