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.
The Concept of Similarity
Similar Triangles
You can think of similar objects as one one being a enlargement or reduction of
the other.
The Concept of Similarity
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
The Concept of Similarity
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?
The Concept of Similarity
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
The Concept of Similarity
Similar Triangles
Two polygons are SIMILAR if they have the same shape but not necessarily of the same size.
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)
The Concept of Similarity
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
The Concept of Similarity
Similar Triangles
T1
T2
These two are similar.
Corresponding angles are congruent
A D
B E
C F
Corresponding sides are proportional:
2DE EF DF
AB BC AC Scale factor from
T1 to T2 is 2
Similar Triangles
The Concept of SimilarityWhich pairs are similar? If they are similar, what is the scale factor?
Similar Triangles
Similar Triangles
Two triangles 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
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
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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
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ABC DEF
.A D
.B E
.C F
Similar Triangles
Similar Triangles
We can also write the
similarity statement as
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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 verticesmatters!!!
Similar Triangles
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ABC DEF
Similar Triangles
A B C D E F
Corresponding
Sides
AB DE
BC EF
AC DF
Proportions from Similar Triangles
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ABC DEF
Similar Triangles
Corresponding
Sides
AB DE
BC EF
AC DF
Proportions from Similar Triangles
Ratios of Corresponding
SidesAB
DE
BC
EF
AC
DF
Suppose
Then the sides of the triangles are proportional, which means:
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.ABC DEF
AB AC BC
DE DF EF
Notice that each ratio consists of correspondingsegments.
Similar Triangles
Proportions from Similar Triangles
The Similarity Statements
Based on the definition of similar triangles, we now have the following SIMILARITY STATEMENTS:
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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.
The Similarity Statements
The Similarity Statements
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
The Similarity Statements
Similar Triangles
Given the triangle similarityLMN ~ 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
The Similarity Statements
In the figure,
Enumerate all the statements that will show that
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.SA ON S A
L
O N
. SAL NOL
Similar Triangles
The Similarity Statements
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
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.SA ON
S A
L
O N
. SAL NOL
Similar Triangles
The Similarity Statements
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
The Similarity Statements
O B L
K
A
Similar TrianglesSimilar Triangles
The Similarity Statements
O L
K
Congruent Angles
KOL ABL
LKO LAB
KLO ALB
Proportional Sides
KO KL OL
AB AL BL
Similar Triangles
Solving for the SidesThe 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
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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
Solving for the SidesNote The ratios can also be formed using any of the following:
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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 orb e c f c f
Given that
If the sides of the triangles are as marked in the figure, find the missing sides.
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A B
C
D E
F
,ABC DEF 68
7
12
Similar Triangles
Solving for the Sides
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A B
C
D E
F
68
7
12 9
DF FE
AC CB
Similar Triangles
Solving for the Sides
Set up the proportions of the corresponding sides using the given sides
For CB:
8 6
12
CB
8 72CB
9CB
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A B
C
D E
F
68
7
12 9
10.5
Similar Triangles
Solving for the Sides
Set up the proportions of the corresponding sides using the given sides
DF DE
AC AB
For AB:
8 7
12
AB
8 84AB
2110.5
2AB or
S A
L
O N
8
10
16
x
y
Similar Triangles
Solving for the Sides
In the figure shown, solve for x and y.
Solution
15
16 8
10
x
For x:
8 160x
20x
8
15 10
yFor y:
10 120y
12y
Check your understanding
The triangles are similar. Solve for x and z.
3 4
12
x
9x
5 4
12
z
15z
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 Proportionality Principles
The Basic Proportionality Theorem
A
D E
B C
A
BC DE
A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality TheoremA
D E
B C
A
BC DE
AD AE DE
AB AC BCProportions:
A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
BC DE AD AE
DB EC
Note The two sides cut by the line segment are also cut proportionally; thus we have
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find the value of x.
Solution
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12 14
x
212
x
24x
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
O B L
K
A
12
6
9
In the figure,
Find OL and OB..KO AB
Solution
12 9
6
OLFor OL:
9 72OL
8OL
For OB:
OB OL BL
8 6
2OB
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if .BC ST
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find BU and SB if
.BC ST
Solution
6
24 12
BUFor BU:
24 72BU
3BU
SB SU BU
For SB:
12 3
9SB
Check your understandingIf , find PQ, PV, and PW.VW QR
22 12
6
PQ
For PQ:
222
PQ
2 22PQ
11PQ
For PV:
11 9 PV
2PV
22
11 2
PW
For PW:
11 44PW
4PW
Similar Triangles
The Proportionality Principles
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
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 15 10
18
x
5 10
6
x
5 60x
12x
Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 21
30 15
x
7
30 5
x
5 210x
42xx
Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any two transversals proportionally.
AB EF CD
and
are transversals.
AC BD
a b
c d
Similar Triangles
The Proportionality Principles
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
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution8
28 16
x
1
28 2
x
2 28x
14x
x
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution6 9
4
x
6 36x
6x
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
3
10 5
x
x10
3
2
5 30x
6x
Check your understandingSolve for the indicated variable.
2. for x and y y
205
28 7
7 5 12
xx
y
1. for a
a15 2510
6 a
a
x
Similar Triangles
The Proportionality Principles
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
The Proportionality Principles
A B
C
D
C D
B
A D
C
A C
B
∆CBD ~ ∆ACD
∆ACD ~ ∆ABC
∆CBD ~ ∆ABC
Similar Right Triangles
Similar Triangles
The Proportionality Principles
A B
C
D
C D
B
A D
C
A C
B
Similar Right Triangles
hab
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
The Proportionality Principles
A B
C
D
hab
y x
c
2 h xy h xy
2 a xc a xc
2 b yc b yc
Similar Right Triangles
This result is also called the GEOMETRIC
MEAN THEOREM for similar right triangles
Similar Triangles
The Proportionality Principles
Similar Right Triangles
The GEOMETRIC MEAN of two positive numbers a and b is
GM abThe geometric mean of 16 and 4 is
16 4GM 64 8
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
36
x
6 3x
18
3 2
Similar Triangles
The Proportionality Principles
Similar Right 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
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
9 25x
3 5
15
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x.
Solution
12 16 x
144 16 x
9x
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x, y and z.
Solution
6 9 x
36 9 x
4x
4 13y
2 13y
9 13z
3 13z
Thank you!