Download MEERA RAVAT

SIMILARITY AND CONGRUENCY
SIMILARITY
SIMILAR FIGURES
Similar figures are figures for which all corresponding angles are
congruent and all corresponding sides are proportional. Example:
There are some figures which are always similar. Example square,
equilateral triangles as they all have same corresponding angles.
SIMILAR TRIANGLES
Triangles that are the same shape but not the same size are said to
be similar.
60º
70º
50º
60º
70º
50º
Corresponding angles: are angles whose vertices correspond in the
one-to-one correspondence.
Corresponding sides: are sides whose endpoints correspond in the
one-to-one correspondence.
If two triangles are congruent, then they must also be similar. But if
two triangles are similar, they are not necessarily congruent. Two
triangles are similar if you can show one of the following conditions.
1. If two triangles have their corresponding angles, then they are
similar (and their corresponding sides are in the same ratio)
2. If two triangles have their corresponding sides in the same
ratio, then they are similar (and their corresponding angles are
equal).
c
b
kc
kb
a
ka
3. If two triangles have a pair of corresponding sides in
proportion and the angle between these sides equal, then they
are similar.
a
x
b
ka
x
kb
Some real life examples of similar polygons can be our exercise
books. They have the same shape (rectangle), but some differ in
sizes, some are big and some are small.
PERIMETERS AND AREAS OF SIMILAR FIGURES
Examples:
1. The rectangles below are similar
3
X
9
6
Find the value of x:
The corresponding sides of similar figures are proportional
Therefore:
9/6=3/x
9x=18
x=2
What is the perimeter of each rectangle?
Perimeter =2L+2W
Perimeter=2L+2W
= 2(3)+2(9)
= 2(2)+2(6)
= 6+18
=4+12
=24
=16
What is the area of each triangle?
Area=L×W
Area= L×W
=3×9
=2×6
=27
=12
2. The isosceles triangles below are similar
6
x
16
8
Find the value of x:
16/8=6/x
16x=48
X=3
What is the area of each triangle?
Area = 1/2×b×h
Area=1/2×b×h
=1/2×16×6
=1/2×8×3
=48
=12
EXERCISE
In this exercise a number written inside a figure represents the
area of the shape in cm². Numbers on the outside give linear
dimensions in cm. In questions 1-4 find the unknown area A. In each
case the shapes are similar.
1.
4cm²
A
3cm
6cm
2.
A
8cm
18cm²
16cm
6cm
3cm
3.
A
27cm²
8cm
12cm
4.
6cm
A
2cm
3cm
5cm²
9cm
Similarity in 3-D shapes
When solid objects are similar, one is an accurate enlargement of
the other.
If two objects are similar and the ratio of corresponding sides as 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 volume has three dimensions, and the scale factor is used three
times.
EXAMPLES
1.
3cm
30cm²
6cm
Two similar cylinders have heights of 3 cm and 6 cm respectively. If
the volume of the smaller cylinder is 30cm³, find the volume of the
larger cylinder.
If linear factor=k, then the ratio of heights (k)=6/3=2
Ratio of volumes (k³)=2³
=8
And the volume of larger cylinder=8×30
=240cm³
2. Two similar spheres made of the same material have weights
of 32kg and 108kg 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.
Ratio of volumes (k³) = 32/108
=8/27
Ratio of corresponding lengths (k) =³√(8/27)
=2/3
Radius of smaller sphere=2/3×9
=6cm
EXERCISE
In this exercise, the objects are similar and a number written
inside a figure represents the volume of the object in cm³.
Numbers on the outside give linear dimensions in cm. In the 2
questions below, find the unknown volume V.
1.
60
V
5
10
2.
5
20
15
V
In the 2 questions below find the length marked by a letter.
1.
6
z
48
9.2
2.
3
24
2
x
CONGRUENCY
Some similar triangles are sometimes called congruent.
Two polygons are congruent if they are the same size and shape that is, if their corresponding angles and sides are equal.
CONGRUENT TRIANGLES
Triangles that are the same shape (all corresponding angles equal)
and the same size (all corresponding sides equal) are said to be
congruent.
Two triangles are congruent if you can show one of the following
conditions.
Side, Side, Side (SSS). If three sides of one triangle are equal to
the three sides of another triangle, then the two triangles are
congruent.
P
Q
X
R
Y
Z
PQR= YZX (SSS)
Side, Angle, Side (SAS). If two sides of one triangle are equal to
two sides of another triangle and the angle between each pair of
sides is the same in both triangles, then two triangles are congruent.
P
E
S
F
G
Q
EFG= SPQ (SAS)
Angle, Side, Angle (ASA) If two angles of one triangle are equal to
two corresponding angles of another triangle and the side between
each pair of angles is the same length in both triangles, then the two
triangles are congruent.
X
Y
XYZ= BCA (ASA)
A
z
B
C
Right angle, Hypotenuse, Side (RHS). If two right-angled triangles
have their longest sides equal in length and another side of the first
triangle is equal to a side of the other triangle, then the two
triangles are congruent.
A
D
B
C
E
F
ABC= FDE (RHS)
A real life example of congruent polygon:
It is used in industries to produce things of same shape and size,
For example books of same size and shape or rulers.
QUESTIONS
In each of the following cases, state whether the two triangles are
congruent.
1.
1
7cm
3cm
3cm
7cm
2.
50º
3cm
5cm
50º
5cm
3cm
GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCY
Writing
skill
3
creativit
y flow
neatnes
s
Drawin
g skill
2
pictures
labeling
Formattin
g skill
3
page
paragraph
s math
equations
Polygon
2
alwayssimilar
polygon
s appln
Triangle
2
theorem
s
applns
2
2
3
0
1
3D
shapes
Questions
3
3
perimete
3D
r
vol,
shapes
SA
congruenc
y variety
0
1
Example
s
2
ex+ans
1
Total
20
10
Your work is incomplete. You have not mentioned about surface area of 3D shapes. You
haven’t given answers for your exercises. Your definitions and interpretations are
mostly wrong.