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The session shall begin
shortly…
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 differ...
The Concept of Similarity
Similar Triangles
You can think of similar objects as one
one being a enlargement or reduction o...
The Concept of Similarity
Similar Triangles
You can think of similar objects as one
being an enlargement or reduction of t...
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 th...
The Concept of Similarity
Similar Triangles
Two polygons are SIMILAR if they
have the same shape but not
necessarily of th...
The Concept of Similarity
Similar Triangles
Q1
Q2
These two are similar.
Corresponding
angles are
congruent
A  E
B  ...
The Concept of Similarity
Similar Triangles
T1
T2
These two are similar.
Corresponding
angles are
congruent
A  D
B  ...
Similar Triangles
The Concept of Similarity
Which 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 ...
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
CONGRUE...
Similar Triangles
 The two triangles shown
are similar because they
have the same three angle
measures.
 The order of th...
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 lette...
 We can also write the
similarity statement as
23
ACB DFE 
 BAC EDF
or CAB FDE 
Similar Triangle Notation
Similar T...
 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
Proportions from Similar Triangles
Ratios of
Corr...
 Suppose
Then the sides of the
triangles are proportional,
which means:
28
.ABC DEF 
AB AC BC
DE DF EF
 
Notice that ...
The Similarity Statements
Based on the definition of
similar triangles, we now
have the following
SIMILARITY STATEMENTS:
2...
30O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the
congruence and
proportionality
statements and
the simil...
The Similarity Statements
31O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionalit...
32O N
E
P
K
I
110
110
30
30
40
40
Similar Triangles
Give the congruence and
proportionality statements
and the simil...
Similar Triangles
Given the triangle similarity
LMN ~ FGH
determine if the given
statement is TRUE or FALSE.
M G   tr...
In the figure,
Enumerate all the
statements that will
show that
34
.SA ON S A
L
O N
. SAL NOL
Similar Triangles
The Simi...
In the figure,
Enumerate all the statements
that will show that
35
.SA ON
S A
L
O N
. SAL NOL
Similar Triangles
The Simi...
Similar Triangles
In the figure,
Enumerate all the
statements that will
show that
.KO AB
. KOL ABL
O B L
K
A
Hint: SEPAR...
O B L
K
A
Similar TrianglesSimilar Triangles
The Similarity Statements
O L
K
Congruent
Angles
  KOL ABL
  LKO LAB
 ...
Similar Triangles
Solving for the Sides
The proportionality of the sides
of similar triangles can be used
to solve for mis...
Similar Triangles
Solving for the Sides
Note The ratios can also be
formed using any of the
following:
39
a
b
c
d
e
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 
6...
41
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...
42
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 us...
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
...
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 th...
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 Theorem
A
D E
B C
A
BC DE
 
AD AE D...
A
B C
D E
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
BC DE 
AD AE
DB EC
Note The ...
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
Find the value of x.
Solution
28
12 14
...
Similar Triangles
The Proportionality Principles
The Basic Proportionality Theorem
O B L
K
A
12
6
9
In the figure,
Find OL...
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 ...
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
...
Similar Triangles
The Proportionality Principles
A bisector of an angle of a triangle
divides the opposite side into segme...
Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 15 10
18

x
5 10
6

x
5 6...
Similar Triangles
The Proportionality Principles
Angle Bisectors
Find the value of x.
Solution 21
30 15

x
7
30 5

x
5 2...
Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any
two transversals proportionally.
...
Similar Triangles
The Proportionality Principles
Three or more parallel lines divide any
two transversals proportionally.
...
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution 8
28 16

x...
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution 6 9
4

x
6...
Similar Triangles
The Proportionality Principles
Parallel Lines and Transversals
Find the value of x.
Solution
3
10 5

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...
Similar Triangles
The Proportionality Principles
The altitude to the hypotenuse of a
right triangle divides the triangle i...
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 Rig...
Similar Triangles
The Proportionality Principles
A B
C
D
C D
B
A D
C
A C
B
Similar Right Triangles
h
ab
y x
c
x
h
h
y
a
b
...
Similar Triangles
The Proportionality Principles
A B
C
D
h
ab
y x
c
2
h xy h xy
2
a xc a xc
2
b yc b yc
Similar Righ...
Similar Triangles
The Proportionality Principles
Similar Right Triangles
The GEOMETRIC MEAN of two
positive numbers a and ...
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
36
x
  6 3x
18...
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find JM, JK and JL.
Solution
8 2
8 2 16 4  JM
...
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find the value of x.
Solution
9 25x
3 5
15
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x.
Solution
12 16 x
144 16 x
9x
Similar Triangles
The Proportionality Principles
Similar Right Triangles
Find x, y and z.
Solution
6 9 x
36 9 x
4x
4 13...
Thankyou!
Math 9   similar triangles intro
Math 9   similar triangles intro
Math 9   similar triangles intro
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Math 9 similar triangles intro

  1. 1. The session shall begin shortly…
  2. 2. Similar Triangles A Mathematics 9 Lecture 3
  3. 3. 4 Similar Triangles What do these pairs of objects have in common? SAME SHAPES BUT DIFFERENT SIZES
  4. 4. 5 Similar Triangles What do these pairs of objects have in common? They are also called SIMILAR objects
  5. 5. The Concept of Similarity Similar Triangles Two objects are called similar if they have the same shape but possibly different sizes.
  6. 6. The Concept of Similarity Similar Triangles You can think of similar objects as one one being a enlargement or reduction of the other.
  7. 7. 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
  8. 8. The Concept of Similarity Similar Triangles Enlargements and Projection
  9. 9. 11 Similar Triangles QUESTION! If a polygon is enlarged or reduced, which part changes and which part remains the same?
  10. 10. 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
  11. 11. 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)
  12. 12. 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
  13. 13. 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: 2 DE EF DF AB BC AC    Scale factor from T1 to T2 is 2
  14. 14. Similar Triangles The Concept of Similarity Which pairs are similar? If they are similar, what is the scale factor?
  15. 15. 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
  16. 16. Similar Triangles Similar Triangles http://wps.pearsoned.com.au/wps/media/objects/7029/7198491/opening/c10.gif
  17. 17. 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
  18. 18. 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
  19. 19. Similar Triangles 21 ABC DEF  Similar Triangles A B C D E F Congruent Angles A D   B E   C F  
  20. 20.  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
  21. 21.  We can also write the similarity statement as 23 ACB DFE   BAC EDF or CAB FDE  Similar Triangle Notation Similar Triangles Why?
  22. 22.  BCA DFE Similar Triangle Notation Similar Triangles  We CANNOT write the similarity statement as  BAC EFD Why?
  23. 23. Kaibigan, sa similar triangles, the correspondence of the vertices matters!!! Similar Triangles
  24. 24. 26 ABC DEF  Similar Triangles A B C D E F Corresponding Sides AB DE BC EF AC DF Proportions from Similar Triangles
  25. 25. 27 ABC DEF  Similar Triangles Corresponding Sides AB DE BC EF AC DF Proportions from Similar Triangles Ratios of Corresponding Sides AB DE BC EF AC DF
  26. 26.  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 Proportions from Similar Triangles
  27. 27. 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
  28. 28. 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
  29. 29. 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
  30. 30. 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
  31. 31. 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 The Similarity Statements
  32. 32. In the figure, Enumerate all the statements that will show that 34 .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!
  33. 33. In the figure, Enumerate all the statements that will show that 35 .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!
  34. 34. 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
  35. 35. 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
  36. 36. 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
  37. 37. Similar Triangles Solving for the Sides 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
  38. 38. 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 Solving for the Sides
  39. 39. 41 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 72CB 9CB
  40. 40. 42 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 84AB 21 10.5 2 AB or
  41. 41. 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: 8 160x 20x 8 15 10  y For y: 10 120y 12y
  42. 42. Check your understanding The triangles are similar. Solve for x and z. 3 4 12  x 9x 5 4 12  z 15z
  43. 43. 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
  44. 44. A B C D E Similar Triangles The Proportionality Principles The Basic Proportionality Theorem A D E B C A BC DE
  45. 45. A B C D E Similar Triangles The Proportionality Principles The Basic Proportionality Theorem A D E B C A BC DE   AD AE DE AB AC BC Proportions:
  46. 46. 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
  47. 47. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem Find the value of x. Solution 28 12 14  x 2 12  x 24x
  48. 48. 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  OL For OL: 9 72OL 8OL For OB:  OB OL BL 8 6  2OB
  49. 49. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem Find BU and SB if .BC ST
  50. 50. Similar Triangles The Proportionality Principles The Basic Proportionality Theorem 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
  51. 51. 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
  52. 52. 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
  53. 53. Similar Triangles The Proportionality Principles Angle Bisectors Find the value of x. Solution 15 10 18  x 5 10 6  x 5 60x 12x
  54. 54. Similar Triangles The Proportionality Principles Angle Bisectors Find the value of x. Solution 21 30 15  x 7 30 5  x 5 210x 42x x
  55. 55. 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
  56. 56. 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
  57. 57. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 8 28 16  x 1 28 2  x 2 28x 14x x
  58. 58. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 6 9 4  x 6 36x 6x
  59. 59. Similar Triangles The Proportionality Principles Parallel Lines and Transversals Find the value of x. Solution 3 10 5  x x 10 3 2 5 30x 6x
  60. 60. 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
  61. 61. 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
  62. 62. 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
  63. 63. Similar Triangles The Proportionality Principles A B C D C D B A D C A C B Similar Right Triangles 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
  64. 64. Similar Triangles The Proportionality Principles A B C D h ab 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
  65. 65. Similar Triangles The Proportionality Principles Similar Right 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
  66. 66. Similar Triangles The Proportionality Principles Similar Right Triangles Find the value of x. Solution 36 x   6 3x 18 3 2
  67. 67. 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
  68. 68. Similar Triangles The Proportionality Principles Similar Right Triangles Find the value of x. Solution 9 25x 3 5 15
  69. 69. Similar Triangles The Proportionality Principles Similar Right Triangles Find x. Solution 12 16 x 144 16 x 9x
  70. 70. Similar Triangles The Proportionality Principles Similar Right 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
  71. 71. Thankyou!
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