chapter 8 similarity
DESCRIPTION
U SING S IMILARITY T HEOREMS. U SING S IMILAR T RIANGLES IN R EAL L IFE. Chapter 8 Similarity. Section 8.5 Proving Triangles are Similar. U SING S IMILARITY T HEOREMS. Postulate. E. D. C. F. B. A. A D and C F . ABC ~ DEF. U SING S IMILARITY T HEOREMS. - PowerPoint PPT PresentationTRANSCRIPT
Chapter 8SimilaritySection 8.5
Proving Triangles are Similar
USING SIMILARITY THEOREMS
USING SIMILAR TRIANGLES IN REAL LIFE
Postulate
A
C
B
D
F
E
A D and C F
ABC ~ DEF
USING SIMILARITY THEOREMS
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
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.
USING SIMILARITY THEOREMS
Compare Side Lengths of LKM and NOP
18
7
62
3
153
5
Ratios Different, triangles not similar
Determine if the triangles are similar
USING SIMILARITY THEOREMS
Compare Side Lengths of LKM and NOP
18 3
30 5
6 3
10 5
15 3
25 5
Ratios Same, triangles are similar
RQS ~ LKM
Determine if the triangles are similar
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
USING SIMILARITY THEOREMS
CED
44°
68°
20
5
2
USING SIMILARITY THEOREMS
Statements Reasons
USING SIMILARITY THEOREMS
Statements Reasons
~
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.
Finding Distance Indirectly
6 2
24x 2 144x 72x
The Tree is 72 feet tall
Finding Distance Indirectly
6 2
24x 2 144x 72x
The Tree is 72 feet tall4
x
724 2
72 x 4 144x 36x
The mirror would need to be placed 36 feet from the tree
HW Pg :6;9;11;13-17;19-25;27-29;32-
34;39-47