BY ALLY ABDALLAH
Similar figures are identical in shape but not in size.
B is an enlargement of A. The lengths have doubled, but the
angles have stayed the same.
NB: - for any pair of similar figures, corresponding angels are in
the same ratio and corresponding angels are equal.
Question:Find the value of ‘a’ and HIJ?
To work out this problem:-
Since we know all the sides are in the ratios are the same ratio
3
𝐹𝐻 𝐴𝐶
𝐼𝐽 𝐷𝐸
𝑎 3
=
18 6
𝑎=
1
= 2= k
6
=
𝑎
1
=
18 2
3×18
6
= 9cm
a=9
We can also use the scale factor method. You take a fraction
which has a numerical number and a denominator. And divide
to get a scale factor ‘k’.
EXAM TIPS: try use the formula which has the ‘variable’ at the
top of the fraction.
As the second part of the question is find HIJ?
As you remember angels are the same among the same figures
so:
CDE = 73
CDE = HIJ
HIJ=73
SIMILAR TRIANGLES:
Triangles are similar when corresponding angles are equal and
their corresponding sides are equal. (Same theory which applies
for the rest of the shapes). Triangles are special in a way
because to prove that two triangles are similar one of the
following statements should be true:
1. The three sides are the same ratio.
2. Two sides are in the same proportion and their included
angles are equal.
3. Corresponding angles are equal
A=X
B=Y
C=Z
QUESTION:By using the statement can you prove ABC and AQR are
similar?
In order to make these questions answerable, draw the
two triangles separately.
Since the angle A remains the same in both triangles, and
we can figure that AC has doubled and corresponded to
AQ while AB has doubled ad corresponded to AR. We can
use statement 2:- two sides are in the same proportion
and their included angels are equal.
CONGRUENCY:
Two plane figures are congruent when one fits into another.
They must be the same size and shape.
A≅E
B≅F
C≅G
D≅H
≅ Means “is congruent to”
Triangles are congruent when one of the four congruency
conditions are observed:
1. Side Side Side ( S.S.S)
2. Side Angle Side( S.A.S)
3. Angle Side Angle (A.S.A)
4. Right Angle Hypotenuse Side (R.H.S)
QUESTION
For the following pairs of triangles state whether they
are congruent or not. If yes give a reason.
Pair one:
Congruent, because of R.H.S rule
Pair two:
Not congruent, because the side of 3cm is not in the same
position on both triangles. Therefore it can’t be A.S.A.
Pair three:
Congruent, because of the S.S.S rule.
AREAS OF SIMILAR FIGURES. (2D
SHAPES)
ABCD∼PQRS
Ratio of corresponding lengths=k
Area of ABCD= l × b
= X× Y=XY
Areas of PQRS= l × b
xk × yk= xyk 2
Ratio of both areas=
𝑥𝑦𝑘2
𝑥𝑦
Ratios of areas=k
Question
Find the ‘unknown area’
Answer
Ratio of corresponding lengths
12
k2 =1.5 ×1.5 =2.25
K= 8
K=1.5
Ratio of corresponding angle=
27
𝐴
27
K2= 𝐴
27
2.25= 𝐴
27
A=2.25
A=12cm2
VOLUME OF SIMILAR OBJECTS. (3D
SHAPES)
The only way 3d shapes can be similar by the volume
A1∼A2
Ratio of corresponding lengths =k
Volume of A1 =a×b×c= abc
Volume of A2=ak× bk ×ck= abck3
𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝐴1
Ratio of volumes=𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝐴2
=
abck3
abc
Ratio of volumes =K3
Question:
Find the” unknown” volume.
54cm3
12
Ratio of corresponding lengths= 8
K=1.5
Ratio of corresponding volumes (K3) =
54
𝑣
54
3.4 = 𝑣
54
V=3.4
V=15.9
THINGS TO REMEMBER IN SIMILARITIES
 Similar figures are identical in shape not in size
 For any pair of similar figures, corresponding sides
are in the same ratio and corresponding angels are
equal.
 Try to use the formula which has the unknown on
top or the fractions while solving one dimension
problems.
 Shapes can be congruent even if one of them is
being rotated or reflected.
 ∼ means ‘similar to’
 ≅ means ‘ congruent to’
𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑛𝑔𝑡ℎ
 Ratio of corresponding lengths=𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑛𝑔𝑡ℎ = k
𝑙𝑎𝑟𝑔𝑒𝑟 𝑎𝑟𝑒𝑎
 Ratio of corresponding angles = 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑎𝑟𝑒𝑎 = k2
𝑙𝑎𝑟𝑔𝑒𝑟 𝑣𝑜𝑙𝑢𝑚𝑒
 Ratio of corresponding volumes = 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑣𝑜𝑙𝑢𝑚𝑒=k3