Math 9 similar triangles intro

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Similar Triangles
A Mathematics 9 Lecture
3

4
Similar Triangles
What do these pairs of
objects have in common?
SAME SHAPES BUT DIFFERENT SIZES

5
Similar Triangles
What do these pairs of
objects have in common?
They are also called SIMILAR objects

The Concept of Similarity
Similar Triangles
Two objects are called similar if they have
the same shape but possibly different
sizes.

Similar Triangles
You can think of similar objects as one
one being a enlargement or reduction of
the other.

Similar Triangles
You can think of similar objects as one
being an enlargement or reduction of the
other (zoom in, zoom out).
The degree of enlargement or reduction is
called the SCALE FACTOR

Similar Triangles
Enlargements and Projection

11
Similar Triangles
QUESTION!
If a polygon is enlarged or reduced,
which part changes and which part
remains the same?

Similar Triangles
Two polygons are SIMILAR if they
have the same shape but not
necessarily of the same size.
Symbol used: ~ (is SIMILAR to)
A C
B
DE
F
In the figure, ABC is
similar to DEF.
Thus ,we write
ABC ~ DEF

Similar Triangles
Two polygons are SIMILAR if they
If they are similar, then
1. The corresponding angles remain
the same (or are CONGRUENT)
2. The corresponding sides are related
by the same scale factor (or, are
PROPORTIONAL)

Similar Triangles
Q1
Q2
These two are similar.
Corresponding
angles are
congruent
A  E
B  F
C  G
D  H
Corresponding sides are
proportional:
1
2
EH EF FG GH
AD AD BC CD
    Scale factor from
Q1 to Q2 is ½

Similar Triangles
T1
T2
These two are similar.
Corresponding
angles are
congruent
A  D
B  E
C  F
Corresponding sides are proportional:
2
DE EF DF
AB BC AC
   Scale factor from
T1 to T2 is 2

Similar Triangles
Which pairs are similar? If they are similar,
what is the scale factor?

Similar Triangles
Similar Triangles
Two triangles are SIMILAR if they
Symbol used: ~ (is SIMILAR to)
A C
B
DE
F
In the figure, ABC is
similar to DEF.
Thus ,we write
ABC ~ DEF

Similar Triangles
Similar Triangles
http://wps.pearsoned.com.au/wps/media/objects/7029/7198491/opening/c10.gif

Similar Triangles
Two triangles are SIMILAR if all of
the following are satisfied:
1. The corresponding
angles are
CONGRUENT.
2. The corresponding
sides are
PROPORTIONAL.
Similar Triangles

Similar Triangles
 The two triangles shown
are similar because they
have the same three angle
measures.
 The order of the letters is
important: corresponding
letters should name
congruent angles.
 For the figure, we write
20
ABC DEF 
Similar Triangles

Similar Triangles
21
ABC DEF 
Similar Triangles
A B C D E F
Congruent Angles
A D  
B E  
C F  

 Let’s stress the order of
the letters again. When we
write note
that the first letters are A
and D, and The
second letters are B and E,
and The third
letters are C and F, and
22
ABC DEF 
.A D  
.B E  
.C F  
Similar Triangles
Similar Triangles

 We can also write the
similarity statement as
23
ACB DFE 
 BAC EDF
or CAB FDE 
Similar Triangle Notation
Similar Triangles
Why?

 BCA DFE
Similar Triangle Notation
Similar Triangles
 We CANNOT write the
similarity statement
as
 BAC EFD
Why?

Kaibigan, sa
similar triangles,
the
correspondence
of the vertices
matters!!!
Similar Triangles

26
ABC DEF 
Similar Triangles
A B C D E F
Corresponding
Sides
AB DE
BC EF
AC DF
Proportions from Similar Triangles

27
ABC DEF 
Similar Triangles
Corresponding
Sides
AB DE
BC EF
AC DF
Ratios of
Corresponding
Sides
AB
DE
BC
EF
AC
DF

 Suppose
Then the sides of the
triangles are proportional,
which means:
28
.ABC DEF 
AB AC BC
DE DF EF
 
Notice that each ratio
consists of corresponding
segments.
Similar Triangles

The Similarity Statements
Based on the definition of
similar triangles, we now
have the following
SIMILARITY STATEMENTS:
29
Congruent
Angles
.A D  
.B E  
.C F  
Proportional
Sides
Similar Triangles
 
AB BC AC
DE EF DF

30O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the
congruence and
proportionality
statements and
the similarity
statement for the
two triangles
shown.

31O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionality statements
and the similarity statement
for the two triangles shown.
Congruent Angles
P O  
I N  
K E  
Corresponding
SidesPI ON
IK NE
PK OE

32O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionality statements
and the similarity statement
for the two triangles shown.
Congruent
Angles
  P I
  I N
  K E
Proportional
Sides  
PI IK PK
ON NE OE
Similarity
Statement  PIK ONE

Similar Triangles
Given the triangle similarity
LMN ~ FGH
determine if the given
statement is TRUE or FALSE.
M G   true
FHG NLM   false
N M   false
LN MN
FG GH
 false
MN LN
GH FH
 true
GF HG
ML NM
 true

In the figure,
Enumerate all the
statements that will
show that
34
.SA ON S A
L
O N
. SAL NOL
Similar Triangles
Note: there is a COMMON
vertex L, so you CANNOT use
single letters for angles!

In the figure,
Enumerate all the statements
that will show that
35
.SA ON
S A
L
O N
. SAL NOL
Similar Triangles
Congruent
Angles
  SAL LON
  ASL LNO
  OLN SLA
Proportional
Sides  
SA AL SL
ON OL NL
Note: there is a COMMON vertex L, so you
CANNOT use single letters for angles!

Similar Triangles
In the figure,
Enumerate all the
statements that will
show that
.KO AB
. KOL ABL
O B L
K
A
Hint: SEPARATE the two right
triangles and determine the
corresponding vertices.
Similar Triangles

O B L
K
A
Similar TrianglesSimilar Triangles
O L
K
Congruent
Angles
  KOL ABL
  LKO LAB
  KLO ALB
Proportional
Sides  
KO KL OL
AB AL BL

Similar Triangles
Solving for the Sides
The proportionality of the sides
of similar triangles can be used
to solve for missing sides of
either triangle. For the two
triangles shown, the statement
38
 
AB BC AC
DE EF DF
can be separated into the THREE
proportions

AB AC
DE DF

BC AC
EF DF

AB BC
DE EF

Similar Triangles
Note The ratios can also be
formed using any of the
following:
39
a
b
c
d
e
f
 
a b c
d e f
 
d e f
a b c
  
a d b e a d
or or
b e c f c f

Given that
If the sides of the
triangles are as marked
in the figure, find the
missing sides.
40
A B
C
D E
F
,ABC DEF 
68
7
12
Similar Triangles

41
A B
C
D E
F
68
7
12 9

DF FE
AC CB
Similar Triangles
Set up the proportions
of the corresponding
sides using the given
sides
For CB:
8 6
12

CB
8 72CB
9CB

42
A B
C
D E
F
68
7
12 9
10.5
Similar Triangles
Set up the proportions
of the corresponding
sides using the given
sides

DF DE
AC AB
For AB:
8 7
12

AB
8 84AB
21
10.5
2
AB or

S A
L
O N
8
10
16
x
y
Similar Triangles
In the figure shown,
solve for x and y.
Solution
15
16 8
10

x
For x:
8 160x
20x
8
15 10

y
For y:
10 120y
12y

Check your understanding
The triangles are similar. Solve for x and z.
3 4
12

x
9x
5 4
12

z
15z

Similar Triangles
The Proportionality Principles
A line parallel to a side of a triangle
cuts off a triangle similar to the
given triangle.
This is also called the BASIC PROPORTIONALITY
THEOREM
BC DE
cuts ABC into
two similar triangles:
DE
~ ADE ABC
A
B C
D E

A
B C
D E
Similar Triangles
The Basic Proportionality Theorem
A
D E
B C
A
BC DE

A
B C
D E
Similar Triangles
A
D E
B C
A
BC DE
 
AD AE DE
AB AC BC
Proportions:

A
B C
D E
Similar Triangles
BC DE 
AD AE
DB EC
Note The two sides cut
by the line segment are
also cut proportionally;
thus we have

Similar Triangles
Find the value of x.
Solution
28
12 14

x
2
12

x
24x

Similar Triangles
O B L
K
A
12
6
9
In the figure,
Find OL and OB.
.KO AB
Solution
12 9
6

OL
For OL:
9 72OL
8OL
For OB:
 OB OL BL
8 6 
2OB

Similar Triangles
Find BU and SB if .BC ST

Similar Triangles
Find BU and SB if
.BC ST
Solution
6
24 12

BUFor BU:
24 72BU
3BU
 SB SU BU
For SB:
12 3 
9SB

Check your
understandingIf , find PQ, PV, and PW.VW QR
22 12
6

PQ
For PQ:
22
2
PQ
2 22PQ
11PQ
For PV:
11 9 PV
2PV
22
11 2

PW
For PW:
11 44PW
4PW

Similar Triangles
A bisector of an angle of a triangle
divides the opposite side into segments
which are proportional to the adjacent
sides.
is the angle
bisector of C.
CD

CB BD
CA DA

Similar Triangles
Angle Bisectors
Solution 15 10
18

x
5 10
6

x
5 60x
12x

Similar Triangles
Angle Bisectors
Solution 21
30 15

x
7
30 5

x
5 210x
42x
x

Similar Triangles
Three or more parallel lines divide any
two transversals proportionally.
AB EF CD
and
are transversals.
AC BD

a b
c d

Similar Triangles
Three or more parallel lines divide any
two transversals proportionally.
Note The cut segment and the length of the
segment themselves are also proportional; thus
we have

 
a c
a b c d

 
b d
a b c d

Similar Triangles
Parallel Lines and Transversals
Solution 8
28 16

x
1
28 2

x
2 28x
14x
x

Similar Triangles
Solution 6 9
4

x
6 36x
6x

Similar Triangles
Solution
3
10 5

x
x
10
3
2
5 30x
6x

Check your
understandingSolve for the indicated variable.
2. for x and y y
20
5
28 7
7 5 12
  
  
x
x
y
1. for a
a15 25
10
6
  a
a
x

Similar Triangles
The altitude to the hypotenuse of a
right triangle divides the triangle into
two triangles that are similar to the
original and each other
A B
C
D

Similar Triangles
A B
C
D
C D
B
A D
C
A C
B
∆CBD ~ ∆ACD
∆ACD ~ ∆ABC
∆CBD ~ ∆ABC
Similar Right Triangles

Similar Triangles
A B
C
D
C D
B
A D
C
A C
B
h
ab
y x
c
x
h
h
y
a
b
a
b
c

h y
x h

a x
c a

b y
c b
Proportions
∆CBD ~ ∆ACD ∆ACD ~ ∆ABC ∆CBD ~ ∆ABC

Similar Triangles
A B
C
D
h
ab
y x
c
2
h xy h xy
2
a xc a xc
2
b yc b yc
This result is also called the GEOMETRIC
MEAN THEOREM for similar right triangles

Similar Triangles
The GEOMETRIC MEAN of two
positive numbers a and b is
GM ab
The geometric mean of 16
and 4 is
16 4GM 64 8

Similar Triangles
Solution
36
x
  6 3x
18
3 2

Similar Triangles
Find JM, JK and JL.
Solution
8 2
8 2 16 4  JM
8 10 80 4 5  JK
2 10 20 2 5  JL

Similar Triangles
Solution
9 25x
3 5
15

Similar Triangles
Find x.
Solution
12 16 x
144 16 x
9x

Similar Triangles
Find x, y and z.
Solution
6 9 x
36 9 x
4x
4 13y
2 13y
9 13z
3 13z

Math 9 similar triangles intro

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Math 9 similar triangles intro