Download Similarity is the position or condition of being similar or possessing

Similarity is the position or condition of being similar or
possessing the same qualities as another object.
Two objects are considered to be similar if they both have the
same shape. Which means, one congruent (when two figures fit
exactly onto each other, they must be the same shape and size)
to the result of a uniform scaling (enlarging or shrinking) of the
other.
Corresponding sides of similar polygons are in proportion, and
corresponding angles of similar polygons have the same
measure. One can be obtained from the other by uniformly
stretching the same amount in all directions, possibly with
additional rotation and reflection i.e., either have the same
shape, or one has the same shape as the mirror image of the
other.
For example, all circles are similar to each other, all squares are
similar to each other, and all parabolas are similar to each other.
On the other hand, ellipses are not all similar to each other, nor
are hyperbolas all similar to each other. If two angles of a
triangle are equal to two angles of another triangle, then the
triangles are similar.
Polygons are considered to be similar not only if their
corresponding angles are equal, the corresponding sides must be
in the same proportion.
Triangles A and B are not similar even though they have the
same angles.
D
C
A
H
B
G
E
F
Similar polygons with more sides. Given any two similar
polygons, corresponding sides are proportional. However,
proportionality of corresponding sides is not sufficient to prove
similarity for polygons beyond triangles (otherwise, for
example, all rhombi would be similar). Corresponding angles
must also be equal in measure.
EXAMPLE 1
1. The corresponding angles are congruent.
2. The corresponding sides are proportional.
Solution: 7/21 = 9/27
1/3 = 1/3.
They are equal therefore the polygons are similar.
Example 2;
1. The corresponding angels are congruent.
Solution; 4/6 = 5/7
0.66 ≠ 0.71
thus;
2. The corresponding sides are proportional.
_In example two, the proportional is not the same thus
they are not similar polygons.
_Conclusion; Similar polygons are polygons for which all
corresponding angles should be congruent and
all corresponding sides should be proportional.
For two shapes to be similar they need to have the same angles
and have to be up to the same scale of one another.
For a triangle, the shape is determined by its angles, so for the
triangles to be similar there has to be a correspondence between
their angles.
E
B
A
C
F
D
So we can say that two triangles
and
if either of the following conditions holds:
are similar
1. Corresponding sides have lengths in the same ratio:
i.e.
.
This is equivalent to saying that one triangle is an
enlargement of the other.
2.
is equal in measure to
, and
measure to
. This also implies that
measure to
.
When two triangles
and
is equal in
is equal in
are similar, we write
Two plane figures are congruent if one fits exactly on to the
other. They must be the same size and shape, but in different
positions (one may be rotated, flipped or placed somewhere
else)
Two triangles are congruent if they satisfy either one of the
following:
(S=SIDE AND A=ANGLE)
1) Two pairs of sides are congruent and the included
angles are congruent. (S.A.S)
2) Two pairs of angles and pair of corresponding sides are
congruent (A.A.S)
3) Three pairs of corresponding sides are congruent
(S.S.S)
4) If two triangles have congruent hypotenuses and one
pair of sides congruent, then the triangles are congruent.
(R.H.S.)
The ratio of perimeters of two similar polygons is
equal to the ratio of the corresponding sides
Similar triangles whose scale factor is 2 : 1.
The ratios of corresponding sides are 4/2, 10/5, 6/3.
These all reduce to 2/1. It is then said that the scale
factor of these two similar triangles is 2 : 1. The
perimeter of
∆ABC is 10 cm, and the perimeter of ∆DEF is 20cm.
When you compare the ratios of the perimeters of these
similar triangles, you also get 2 : 1.
If two similar triangles have a scale factor of a: b, then
the ratio of their perimeters is a :b.
1. When two triangles are similar, the reduced ratio of any
two corresponding sides is the called
The Scale Factor of the similar triangles; a: b
2. If two similar triangles have a scale factor of a: b then
the ratio of their perimeter is a: b.
3. If two similar triangles have a scale factor of a: b, then
the ratio of their areas is a2: b2
B
Example 1;
1.
10
J
4
6
5
Similarity Ratio=1/2
Ratio of Perimeter;
A
8
Perimeter of triangle ABC= 10+6+8
C
H
3
=24
Perimeter of triangle HJK= 5+3+4
=12
Therefore: Perimeter of triangle HJK/ Perimeter of triangle
ABC
= 12/24
= 1/2
Ratio of areas= (1/2)2=1/4
K
Ratio of surface areas of similar figures; if the ratio of
corresponding sides of figure A and figure B is
a : b, then the ratio of their corresponding surface is a2 :
b2.
Ratio of volumes of similar figures; if the ratio of
corresponding sides of figure A and B is a : b, then
the ratio of their corresponding volume is a3 : b3.
The surface areas of two similar regions are proportional
in the ratio k2 when k is the line ratio of the
regions.
The volumes of two similar solids are proportional in the
ratio k3 when k is the line ratio of the solids.
Example 1;
Volume of A= 2x3x4 Volume of B= 4x6x8
= 24cm3 =192cm3
Ratio of volume= Volume of cube B/volume of cube A
Ratio of volumes=192/24
=8
=23
Therefore; Ratio of volume: (ratio of sides)3