A scale factor is a factor by which all the components of an object multiplied in order to create proportional enlargement or reduction.
If the scale factor of two similar figures is a/b, then
Example 1 :
Two similar solids have side lengths in the ratio 2 : 5.
(a) What is the ratio of their volumes ?
(b) If the smaller shape has a volume of 100 mm^{3}, what is the shape of the larger shape ?
Solution :
(a) If two shapes are in the ratio a : b, then ratio of their volumes is a^{3} : b^{3}.
Ratio of its volume = 2^{3} : 5^{3}
= 8 : 125
(b) Volume of the smaller shape = 100 mm^{3}
Ratio of their volume = (Ratio of the sides)^{3}
Volume of smaller shape/Volume of larger shape = (2/5)^{3}
100/Volume of larger shape = 8/125
Volume of larger shape = 100(125)/8
= 1562.5 cm^{3}
Example 2 :
Two similar solids have volumes of 20 m^{3} and 1280 m^{3}. James says that the sides of the larger solids are 4 times as long as the sides of the smaller shape. Claire says that the sides are 8 times longer. Who is correct ?
Solution :
Ratio of volume of solids = 20/1280
= 1/64
(a/b)^{3} = (1/64)
(a/b)^{3} = (1/4)^{3}
Volume of smaller shape / Volume of larger shape = 1/64
Volume of Larger shape = 4 x Volume of smaller shape
So, James is correct.
Example 3 :
Find the values of x and y and the measure the angle P.
Solution :
TS/QP = TV/QR = SV/PR
4/6 = 7/y = x/9
2/3 = 7/y = x/9
2/3 = 7/y 2y = 21 y = 21/2 y = 10.5 cm |
2/3 = x/9 18 = 3x x = 18/3 x = 6 |
In the similar figures, the side length may change the angle measure does not.
So angle P is also 86 degree.
Example 4 :
Two polygons are similar. If the ratio of their perimeters is 3 : 4, find the ratio of the corresponding sides.
Solution :
Ratio of the corresponding sides = Ratio of their perimeter
Ratio of the corresponding sides = 3 : 4
Example 5 :
Two polygons are similar. If the ratio of the area is 100 : 49, find the ratio of their corresponding sides.
Solution :
Ratio of areas of similar figures
= (Ratio of the corresponding sides)^{2}
Let a and b be the corresponding sides.
100 : 49 = (a : b)^{2}
10 : 7 = a : b
Ratio of corresponding sides is 10 : 7.
Example 6 :
Two triangles are similar and the perimeter of the smaller one is 50. The ratio of the corresponding sides is 3 : 4, find the perimeter of the larger triangle.
Solution :
Ratio of corresponding sides = Ratio of their perimeter
3 : 4 = Perimeter of smaller triangle : perimeter of larger triangle
3/4 = Perimeter of smaller triangle/Perimeter of the larger
3/4 = 50/Perimeter of larger triangle
Perimeter of larger triangle = 50(4)/3
= 66.67
Example 7 :
2 triangles are similar and a side of the smaller one is 8. The ratio of the corresponding areas is 25:36, find the length of the corresponding side of the larger triangle.
Solution :
(Ratio of corresponding side)^{2} = Ratio of their areas
Let a and b be the corresponding sides.
(a:b)^{2} = 25 : 36
a:b = √(25:36)
Ratio of the corresponding sides is 5 : 6.
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