Download RAVINA PATTNI 10B SIMILARITY

SIMILARITY
Similar Polygons.
Note that the two polygons to the left differ in size but are
alike in shape.The two polygons are said to be similar.
A formal definition of similar (~) polygons includes terms such
as one-to-one correspondence, corresponding angles and corresponding sides.
Two polygons are said to be similar if and only if there is a one-to-one correspondence
between their vertices such that:
1. Corresponding angles are congruent (equal ())
2. Lengths of corresponding sides are in proportion.
The polygons shown are similar. When we say polygon CDEF is similar to polygon
VWXY, we are asserting that the vertices have been paired as follows:
C
V
D
W
E
X
F
Y
Y
1. Angle C  angle V
F
E
Angle D  angle W
V
Angle E  angle X
Angle F  angle Y
C
X
W
D
2.
CD DE EF FC



VW WX XY YV
At this stage if we wish to show that two polygons are similar, we must establish that
both conditions of the definition are met. The following figures show that meeting just
one condition is not sufficient.
5
5
4
4
3
5
4
4
Polygon 1
Polygon 2
Polygon 3
Notice that although the corresponding angles of polygons 1 and 2 are congruent, the
lengths of corresponding sides are not in proportion.
Now look at polygons 2 and 3. In this case the lengths of the corresponding sides are in
proportion but the corresponding angles are not congruent. Again the polygons are not
similar.
1
R
6
W
3
A
4
12
6
D
8
S
2
T
B
4
C
The perimeter of a polygon is the sum of the lengths of its sides. Consider the perimeters
of the similar quadrilaterals above. Note that the ratio if the perimeter of the quadrilateral
RSTW to the perimeter of quadrilateral ABCD is 15:30 or 1:2. How does this ratio
compare with the ratio of the lengths of each pair of corresponding side?
1. If two polygons are similar, the ratio of their perimeter equals to the ratio of the
lengths of any pair of corresponding sides.
Given: Polygon MORST is similar to polygon M’O’R’S’T’, the polygons have
perimeter p and p’ respectively.
T’
T
M
S
M’
S’
O
R
O’

R’
p
MO

p' M ' O'
Exercise 1
Find the unknown sides for each of the following:
1.
C
4
O
y
x
5
B
N
P
2
3
z
D
M
6
C
2
2.
T
10
y
S
C
4
x
6
B
3
A
W
z
R
D
7
3.
I
8
y
E
6
H
D
z
5
3
F
J
K
G
x
6
4.
P
x
O
D
8
3
C
4
M
5
5
N
2
A
y
B
5. The drawing shows a rectangular picture 16cm × 8cm surrounded by a border of width
4cm. are the two rectangles similar?
16
8
3
6. The diagonals of a trapezium ABCD intersect at O. AB is parallel to DC, AB = 3cm
and DC = 6cm. If CO – 4cm and OB = 3cm, find AO and DO.
7. From the rectangle ABCD a square is cut off to leave rectangle BCEF.
Rectangle BCEF is similar to ABCD. Find x and hence state the ratio of the sides of
rectangle ABCD. ABCD is called the Golden Rectangle and is an important shape in
architecture.
A
F
B
1
D
1
E
C
Similar triangles.
At this point in our discussion of similar polygons, the only way that we can rove two
triangles similar is by showing that the triangles satisfy the definition of similar polygons.
1. If two angles in one triangle are congruent to two angles of another triangle, the
triangles are similar (AA)
Example:
B
D1
E 4
3
2
C
F
Given: Plane figure with angle 1  angle 2
Angle 3  angle 4 because they are vertically opposite angles.
DE DF

Because lengths of corresponding sides of similar triangles are in
CE CB
proportion.
 DEF ~  CED
2. If an angle of one triangle is congruent to an angle of another triangle and the
lengths of the sides including these angles are proportional, the triangles are similar.
C
T
A
B
R
Given:  ABC and  RST with angle C  angle T
AC BC

RT
ST
  ABC ~  RST
S
4
Example:
A tree of height 4m casts a shadow of length 6.5cm. Find the height of a house casting a
shadow 26m long.
Tree
House
6.5m
26m
4m
x
6.5 4

26 x
26 × 4 = 6.5x
104 6.5 x

x = 16
6.5 6.5
Therefore, the height of the house is 16m.
Exercise 2
State whether the triangles are similar
1,
500
30
1000
0
2.
300
700
5
3.
300
a
a
b
b
700
700
R
4. Complete the following question:
a) Triangle RST ~ Triangle _______
b) Complete the extended proportion


RS ST RT
c) If RM = 3, MN = 4 and PS = 7, then ST = _______
d) If RM = 4, MN = 5 and ST = 8, then RS = _______
1
M
N
2
S
T
5. Find the sides marked with letters in the following question.
A
Q
x
16cm
y
6cm
B
C P
3cm R
6cm
3. If a line is parallel to one side of a triangle and intersects the other two side, it
divides them proportionally.
Given: Triangle ABC;
C
Line YZ is parallel to line AB
Angle 1  angle 2;
Y 2
4 Z
Angle 3  angle 4
AY BZ

1
3
YC ZC
A
B
6
Example:
Given  ABC with line YZ parallel to line AB
Find the ratio of:
a) CZ to ZB
CZ CY 7


ZB YA 4
C
7
Y
4
b) BC to ZC
BC AC 11


ZC YC
7
A
Z
B
c) BC to BZ
BC AC 11


BZ AY 4
Exercise 3
Find the sides marked with letters in questions 1 to 3; all lengths are given in centimeters.
1.
3.
C
X
w
3
4
B
6
B
y
D
3
4
2
A
C
A
x
E 2
D
10
2.
P
m
Q
2
R
5
3
T
3
S
Summary:
1. If two convex polygons are similar:
a) Corresponding angles are congruent
b) Lengths of corresponding sides are in proportion
c) The ratio of the perimeters equals the ratio of the lengths of any pair of corresponding
sides.
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2. Two triangles are similar if two angles of one triangle are congruent to tow angles of
the other triangle.
3. In any triangle, a line that is parallel to one side and intersects the other two sides
divides them proportionally.
4. It is important to note that some figures are a must to be similar. These figures include:
 Two equilateral triangles
 Two squares
 Two regular pentagons
 Two circles
Areas of similar shapes
The two rectangles are similar, the ratio of the corresponding sides
being k.
A
B
W
area of ABCD = ab
a
area of WXYZ = ka × kb = k2ab
ka
2
2
 Area WXYZ = k ab = k
C
b
D
Area ABCD
ab
Z
This illustrates an important general rule for all similar shapes:
X
kb
Y
If two figures are similar and the ratio of corresponding sides k, then the ratio of their
areas is k2
Note: k is sometimes called the linear scale factor
This result also applies for the surface area of similar three dimensional objects.
Area scale factor = (linear scale factor)2
Example 1:
XY is parallel to BC.
AB = 3
AX 2
If the area of triangle AXY = 4cm2, find the area of triangle ABC. 3 2
The triangles ABC and AXY are similar.
X
Area scale factor = (linear scale factor) 2
4 = 22
B
2
x 3
x=9
A
Y
C
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Example 2:
Two similar triangles have areas of 18cm2 and 32cm2
respectively. If the base of the smaller triangle is 6cm,
find the base of the larger triangle.
Area scale factor = (linear scale factor)
18 = 62
32 x2
x = 8cm
18cm
32cm
m
6cm
x cm
Example 3:
A floor is covered by 600 tiles which are 10cm by 10cm. How many 20cm by 20cm tiles
are needed to cover the same floor?
Total area = 10 × 10 × 600
= 60000cm2
 For 20cm by 20cm, 60000
20×20
Therefore, 150 tiles are needed to cover the same floor.
Exercise 4:
1. Find the unknown area A. In each case the shapes are similar.
a)
4cm2
A
3cm
6cm
b)
2cm
3cm2
6cm
A
c)
A
8cm
16cm
18cm2
16cm
3cm
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d)
A
27cm2
8cm
12cm
2. Find the lengths marked for each pair of similar shapes.
a)
5cm2
4cm
20cm2
x
b)
4cm2
9cm2
3cm
z
c)
12cm2
5cm
3cm2 a
3. Given Ad = 3cm, AB = 5cm and area of triangle ADE = 6cm2
Find:
a) Area of triangle ABC
b) Area of DECB
A
D
E
B
4. The triangles ABC and EBD are similar (AC and DE are not
parallel)
If AB = 8cm, BE = 4cm and the area of triangle DBE = 6cm2, find
the Area of triangle ABC
C
A
D
B
E
C
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5. A wall is covered by 160 tiles which are 15cm by 15cm. how many 10cm by 10cm tiles
are needed to cover the same wall?
6. When potatoes are peeled do you lose more peel or less when big potatoes are used as
opposed to small ones?
Congruency
Suppose we match the vertices of triangle ABC with those of triangle DEF in the following
way
A
D
B
E
C
F
This enables us to speak of correspondence between the triangles:
Triangle ABC
Triangle DEF
C
F
A
B
D
C
In this correspondence, the first vertices named A and D, are corresponding vertices. So are
the second and the third vertices named. Because A and D are corresponding vertices, angle
A and angle D are called corresponding angles of the triangles. Other corresponding angles
are angle B and angle E; angle C and angle F.
Because vertices A and B correspond to vertices D and E, line AB and line DE are called
corresponding sides. Other corresponding sides are line BC and line EF; line AC and line
DF.
When the above six statements are true for triangle ABC and triangle DEF, the triangles
are said to be congruent ( ) triangles.
There are several ways to find if triangles are congruent. These include:
1. If three sides of one triangle are congruent to the corresponding parts of another
triangle, the triangles are congruent. (SSS)
C
F
A
B
D
E
According to postulate:
If line AB  line DE, line BC  line EF, and line AC  line DF then
 ABC is   DEF.
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2. If two sides and the included angle of one triangle are congruent to the
corresponding parts on another triangle, the triangles are congruent. (SAS)
C
F
A
B
D
E
According to postulate:
If line AB  line DE, line AC  line DF and
angle A  angle D then  ABC is congruent to  DEF
Exercise 5
1. Fill in the blanks:
a) Pair vertex A with vertex ______
b) Pair vertex C with vertex ______
c) Angle B and angle ______ are corresponding angles.
d) Line CB and line _____ are corresponding sides.
e) Which statement is correct,  ABC   KSV or  ABC   KVS?
15
10
C
B
7
10
K
S
7
A
15
V
2. Prove two triangles are congruent.
a) Pair vertex D with vertex ______, and vertex C with vertex ______
b) Line DF and ______ are corresponding sides.
c) Line EF and ______ are corresponding sides.
d) Is the statement,  DEF   ZJO correct?
e) Is the statement,  OZJ   FDE correct?
F
D
O
1100
1100
E
Z
J
12
Some ways to find right triangles that are congruent are:
1. If two legs of one right triangle are congruent to the corresponding parts of
another right triangle, the triangles are congruent. (LL)
T
R
S
Z
X
Y
 RST and  XYZ;
Angle S and angle Y are right angles.
Line RS  line XY; line ST  line YZ
  RST   XYZ.
2. If the hypotenuse and a leg of one right triangle are congruent to the
corresponding parts of another right triangle, the triangles are congruent. (HL)
According to postulate:
If triangles RST and XYZ are right triangles with right angles S and Y, line RT  XZ
and line RS  line XY then  RST   XYZ.
Exercise 6
1. In the figure it is given that line XC  line AE, line AX  line XD and line BX  line
XE. Name a right triangle that has:
a) Hypotenuse XD
X
b) Hypotenuse XE
c) Line XD as one of its legs
d) Line BX as one of its legs
e) Line AS as its hypotenuse.
f) Name every right triangle that has XC
as one of its legs.
A
B
C
D
E
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2. Name each right triangle shown in the figure:
a) Given: Angle ACD is a right angle; line CD is perpendicular to line AB
C
A
D
B
b) Given: Angle ABC, angle BCD, angle CDA and angle DAB are right angles.
D
C
E
A
B
More ways to find triangles congruent are:
1. If two angles and the included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent. (ASA)
C
A
F
B
D
E
According to postulate:
If angle A  angle D, line AB  DE, and angle B  angle E; then  ABC   DEF.
14
2. If two angles and a not- included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent. (AAS)
C
F
D
A
E
B
According to theorem:
Triangles ABC and DEF;
Angle A  angle D; angle C  angle F; line AB  line DE.
 ABC   DEF.
Exercise 7
1. Name the side that is included between the angles named:
a) A, B
b) B, C
A
c) A, C
C
B
2. Name the two sides that are not included between the angles named:
a) D, E
D
b) E, F
c) D, F
F
E
More ways to find right triangles congruent are:
1. If the hypotenuse and an acute angle of one right triangle are congruent to the
corresponding parts of another right triangle, the triangles are congruent. (HA)
15
According to theorem:
Triangles ABC and DEF; angle B and angle E are
right angles; line AC  line DF;
angle A  angle D
  ABC is congruent to  DEF
C
A
B
F
D
E
2. If a leg and an acute angle of one right triangle are congruent to the
corresponding parts of another right triangle, the triangles are congruent.
Case 1
Triangles RST and XYZ; angle S and angle Y are
right angles; line RS  line XY; angle
R  angle X
T
Z
  RST   XYZ.
R
S X
Y
Z
Case 2
Triangles RST and XYZ; angle S and angle Y are
right angles; line RS  line XY; angle
T  angle Z.
  RST   XYZ.
T
X
R
Y
S
16
Exercise 8
State whether the HA theorem, the LA theorem, or some other right triangle method can
be used to prove that the triangles are congruent.
1.
2.
3.
4.
17
5.
Summary
1. Ways to prove two triangles congruent:
All triangles
Right triangles
SSS ASA
LL HA
SAS AAS
HL LA
2. A common way to prove that two angles or two segments are congruent is to show that
they are corresponding parts of congruent triangles.
3. If two sides of a triangle are congruent, the angles opposite those sides are congruent.
4. If two angles of a triangle ate congruent, the sides opposite those angles are congruent.
Similar 3D shapes
Volume of similar objects:
When solid objects are similar, one is an accurate enlargement of the other. If two objects
are similar and the ratio of corresponding sides is k, then the ratio of their volumes is k3.
A line has one dimension, and the scale factor is used once.
An area has two dimensions, and the scale factor is used twice.
A surface area of 3 dimensions figures also uses the scale factor twice.
A volume has three dimensions, and the scale factor is used three times.
Volume scale factor = (linear scale factor)3
18
Example:
1.
6cm
3cm
30cm3
Two similar cylinders have heights of 3cm and 6cm respectively. If the volume of the
smaller cylinder is 30cm3 , find the volume of the larger cylinder.
If the linear scale factor = k, then ratio of heights (k) =

6
=2
3
ratio of volumes (k3) = 23
=8
And volume of larger cylinder = 8 × 30
= 240cm3
2. Two similar spheres made of the same material have weights of 32kg and 108 kg
respectively. If the radius of the larger sphere is 9cm, find the radius of the smaller
sphere.
We may take the ratio of weights to be the same as the ratio of volumes.
32
108
8
=
27
Ratio of volumes (k3) =
Ratio of corresponding lengths (k) =
3
8
27
2
3
2
 Radius of smaller sphere =
×9
3
= 6cm
=
19
Exercise 9
In this exercise, the objects are similar and a umber written inside a figure represents the
volume of the object in cm3.
Numbers on the outside give linear dimensions in cm. Find the unknown volume, V for
questions 1- 4
1.
60
V
5
10
2.
4.5
V
Radius = 1.2
Radius = 12cm
3.
V
54
8
12
4.
88
V
6.1
3.1
20
In questions 5 and 6, find the lengths marked by a letter.
5.
54
16
6
m
6.
7 10
y
270
7. Two spherical balls are made with the same material. Their masses are 30kg and 18kg
respectively. If the radius of the smaller sphere is 3cm, find that of the larger one.
8. Two similar cylindrical tins have base radii of 6cm and 8cm respectively. If the
capacity of the larger tine is 252cm3, find the capacity of the small tin.
9. Two similar cones have surface areas in the ratio 4:9. Find the ratio of:
a) their lengths,
b) their volumes.
10. A container has a surface area of 5000cm2 and the capacity is 12.8 litres. Find the
surface area of a similar container which has a capacity of 5.4 litres.
21
Answers:
Exercise 1
1
2
1. x = 6 , y = 3 , z = 4
3
3
4
1
4. x = 4 , y = 8
5
5
7. 0.618 ; 1.618 :1
2. x = 5, y = 6
2
2
, z = 11
3
3
1
1
2
3. x = 5 , y = 6 , z = 2
3
5
2
5. No
6. 2cm, 5cm
1. Yes
2. No
3. No
4. (a) RMN
(b) RM, MN, RN
(c) 13
Exercise 2
(d) 6
2
5
1
3
5. x = 12cm, y = 8cm
Exercise 3
1. y = 6
1
4
2. m = 5 , z = 4
3
5
3. x = 4, w = 1
1
2
Exercise 4
1. (a) 16cm2
(d) 12cm2
1
cm
2
4. 24cm 2
(c) 2
(b) 27cm2
2. (a) 8cm
3. (a) 16
2
cm 2
3
5. 360
(c) 128cm2
1
(b) 4 cm
2
2
(b) 10 cm2
3
6. Less (for the same weight)
Exercise 5
1. (a) K
(d) line SV
(b) line OZ
(e) Yes
Exercise 6
1. (a)  XCD
(d)  BXE
(b) S
(e)  ABC   KVS
(c) line OJ
(c) V
2. (a) Z, J
(d) Yes
(b)  XCE
(e)  AXD
(c)  AXD
(f)  ACX,  XCE,
 XCD,  XCB
2. (a)  ACB,  CDB,  ADC (b)  ABC,  BCD,  CDA,
 AEB,  DAB,  DEC,
 BEC,  AED
22
Exercise 7
1. (a) line AB
2. (a) line DF, EF
(b) line BC
(b) line DE, DF
(c) line AC
(c) line DE, EF
2. AA
5. HA
3. LA
Exercise 8
1. LA
4. HA
Ravina, you have done well overall..but could have given more examples and
some real life problems…
GRADE X - ASSESSMENT - SIMILARITY AND
CONGRUENCY
Writing Drawin
skill
g skill
3
2
creativit picture
y flow
s
neatnes labelin
s
g
3
2
Formatti
ng skill
3
page
paragrap
hs math
equations
3
Polygo
n
2
alwayssimilar
polygo
ns
appln
1
Triangle
2
theore
ms
applns
3D
shapes
3
perimet
er
vol, SA
2
3
Exampl
Questions es
3
2
3D
ex+ans
shapes
congruen
cy
variety
1
1
Total
20
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