Download chap-15-similar-shapes

Similarity
Two figures are similar if 1 of these conditions is satisfied
1. All three angles are equal
2. the corresponding sides are in the same ratio
Therefore AB = BC = AC
XY YZ XZ
1. All angles are equal
Example a)
A = E ( given)
B = G ( given)
C = F ( given)
Therefore triangles ABC are similar
EGF
Important: When you write down the final therefore……, the letters have to match so write the
letters in the order that you have proved them
Example b)
Page 1 of 4
Angle ABC = angle EDC
( alternate angles)
Angle BAC = angle DEC
(alternate angles)
Angle ACB = angle ECD
( vertically opposite angles)
Therefore triangles ABC are similar
EDC
Example c)
If DE and AC are parallel
Angle B is common
BDE = BAC ( corresponding angles)
BED = BCA ( corresponding angles)
Therefore triangles BDE are similar
BCA
Once the triangles have been proven to be similar, a missing side can be found.
Example : Consider the above diagram in which BD = 4cm, BA = 9cm, BE = 3cm.
Find BC
Since the traiangles are similar
BD = DE = BE
BC
CA
BA
One of the ratios is not needed so the values are placed in the other two.
4 = 3
9
BC
flip the ratios to put BC on top
9 = BC
4
3
Therefore
BC = 3 x 9
4
Page 2 of 4
Reminder : Two shapes are said to be similar if their sides are in the same
ratio
Example : Consider these two triangles. The large triangle has a base of 3cm
and a height of 6cm. The small triangle has a base of 2cm and a height of
4cm
Comparing bases : base of large triangle = 3 = 3
Base of small triangle 2
2
Comparing heights : height of large triangle = 6 = 3
Height of small triangle
4
2
The value obtained by comparing linear measurements is called the scale
factor,k
Therefore, in this case the scale factor is 3 : 2
Areas
Consider the area of the large triangle
A = ½ bh = ½ x 3 x 6 = 9cm2.
Now the area of the small triangle
A = ½ bh = ½ x 2x 4 = 4cm2.
Lets find the area factor
Area of large triangle = 9
Area of small triangle 4
Therefore in this case the area factor is 9 : 4
Given the scale factor, the area factor is simply found by squaring both
numbers
In general. If the scale factor is x, then the area factor is x2
y
y2
Page 3 of 4
Volumes
Volumes work out in pretty much the same way.
Given the scale factor, the volume factor is simply found by cubing both
numbers
In general, if the scale factor is x , then the volume factor is x3
y
y3
Page 4 of 4