SIMILARITY AND CONGRUENCY.
Two figures (e.g. polygons) are similar when they both have the same shape, and one object is congruent (equal) to the uniform enlargement of the other. The corresponding (i.e. equivalent) sides of similar polygons are in proportion, and corresponding angles of similar polygons are equal in measure. In short, this means that the figures have the same shape but not the same size.
Below, the shapes shown in the same colour are similar:
The rectangles below are similar to each other:
Many times, you will be asked to find the angles and sides of figures. Similar polygons can help you out:
Find the value of x, y, and the measure of angle P.
Solution:
To find the value of x and y, write proportions involving corresponding sides. Then use cross multiplication to solve.
x = 6 y = 10.5
To find angle P, note that angle P And angle S are corresponding angles. By definition of similar polygons, angle P = angle S = 86o.
SPECIAL SIMILARITY RULES FOR TRIANGLES.
The triangles have a couple of special rules dealing with similarity. They are:
1) Prove triangle ABE is similar to triangle CDE.
Solution:
Angle A and angle C are congruent (this information is given in the figure).
Angle AEB and angle CED are congruent because vertical angles are congruent. Triangle ABE and triangle CDE are similar by Angle-Angle.
.
2) Are the triangles shown in the figure similar?
Solution:
Find the ratios of the corresponding sides.
The sides that include angle V and angle L are proportional.
Angle V and angle L are congruent (the information is given in the figure).
Triangle UVS and triangle KLM are similar by Side-Angle-Side.
NOTE: Some figures will always remain similar. These are squares, circles and equilateral triangles. The reason for them to remain similar is that the ratio of the sides is always 1 (e.g. in squares the ratio of the sides is always 1:1:1:1)
EXERCISE 1:
The following figures are similar. Find the sides marked with letters. (in cm)
1) 2
a 5
e
4
6
2) 9
x 8 12 15
y
3)
x 16 y 6
6 3
AREAS OF SIMILAR SHAPES
The two rectangles below are similar, and the ratio of the corresponding sides is ‘k’.
(NOTE: ‘k’ is sometimes called the scale factor)
W X
A B
A ka
D b C
Z Y
kb
Area of ABCD= ab
Area of WXYZ=k2ab
= k2
This shows an important rule for all similar figures: if two figures are similar and their ratio is ‘k’, then the ratio of their areas will be ‘k2’
A
XY is parallel to BC.
2
3
X Y
B C
If the area of triangle AXY=4cm2, find the area of triangle ABC.
Solution:
Triangle ABC and AXY are similar.
The ratio of corresponding sides (k) =
Ratio of areas (k2) =
Area of triangle ABC= × (area of AXY)
= × 4 =9 cm2
CONGRUENT FIGURES
Two-dimensional figures are congruent only if one of the figures fits exactly on the other. This means that they must be the same shape and size.
The following shapes are congruent:
CONDITIONS FOR CONGRUENT FIGURES.
Two triangles are congruent if you can show one of the following conditions:
NOTE: the longest side of a right-angled triangle is always opposite the right angle. This side is known as the hypotenuse of the triangle.
EXERCISE 2:
In each of the following cases, state whether the two triangles are congruent and, if they are, give a reason. (the diagrams are not to scale.)
a)
85º
45º
3 cm
45º 85º
3 cm
b)
50º
3 cm 3 cm
50º
5 cm 5 cm
c)
80º
3 cm 3 cm
60º 40º
4 cm 4 cm
d) 3 cm
7 cm
7 cm
3 cm
3-D SHAPES.
· 3-D shapes are similar when one shape is an exact enlargement of the other.
· A line has one dimension, and so the scale factor (k) is used once
· An area has two dimensions, and so the scale factor is used twice (k2)
· A volume has three dimensions, and so the scale factor is used three times (k3)
Two similar cylinders have heights of 3cm and 6cm respectively. If the volume of the smaller cylinder is 30 cm3, find the volume of the larger cylinder.
3 cm 30cm2 6 cm
Solution:
Ratio of heights (k) = = 2
Ratio of volume (k3) = 23
= 8
Volume of large cylinder = 8×30
= 240 cm3
Two similar spheres made of the same material have weights 32kg and 108kg respectively. If the radius of the larger sphere is 9cm, find the radius of the smaller sphere.
Solution:
We may take the ratio of weights to be the same as the ratio of volumes.
Ratio of volumes (k3) =
=
Ratio of corresponding lengths (k) =
=
Radius of smaller sphere = × 9
= 6
EXERCISE 3:
In this exercise, all the objects are similar and the number written inside the figure represents the volume of the objects (in cm3)
The numbers on the outside give the dimensions (in cm)
Find the volume:
1)
5 cm
10 cm
2)
5 cm 20 15 cm
V
3)
4.5
V
Radius = 1.2 cm
Radius = 12 cm
4)
8 cm 12 cm
ANSWERS:
Exercise 1:
1) a= 2 , e= 3 cm
2) x= 6cm, y= 10 cm
3) x= 12 cm, y= 8 cm
Exercise 2:
‘a’, ‘c’ and‘d’ are correct. ‘a’ is correct because it follows the ‘AA’ similarity. (Angle-Angle similarity). ‘c’ is correct because it follows the ‘SAS’ similarity (Side-Angle-Side similarity). ‘d’ is correct because follows the ‘RHS’ similarity (Right angle, Hypotenuse, Side similarity)
Exercise 3:
1) 480 cm3
2) 540 cm3
3) 4500 cm3
4) 16 cm3
Shaista, read the instructions properly.. you haven’t discussed about surface area, perimeter and similarity of different polygons.
GRADE X - ASSESSMENT - SIMILARITY AND CONGRUENCYWriting skill / Drawing skill / Formatting skill / Polygon / Triangle / 3D shapes / Questions / Examples / Total
3 / 2 / 3 / 2 / 2 / 3 / 3 / 2 / 20
creativity flow neatness / pictures labeling / page paragraphs math equations / always-similar polygons appln / theorems applns / perimeter vol, SA / 3D shapes congruency variety / ex+ans
2 / 2 / 3 / 1 / 2 / 1 / 2 / 1 / 14