Similar Figures

Similar Figures are those figures which have the same shape, but the magnitude of their dimensions may or may not be equal. If the magnitude of their dimensions is also equal, then they are said to be congruent figures. All congruent figures are similar, but all similar figures are not congruent.


Some geometric forms are always identical in design. Imagine a circle, although the size of the object tends to change, the form stays the same. It may be assumed, thus, that all circles of varying radii are similar to each other. The figure shown below indicates the concentric circles whose radii are different, but all of them are identical. However, since their sizes are different, they are not congruent.


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Similar Figures Real Life Examples

For example, in real life, the front wheels of a vehicle, the hands of a human, two teacups, etc. are representations of congruent figures or objects. All identical shape items have the same form, but the measurements are different. The ∼ sign is used to symbolize similarity.


Scale Ratio

For two similar figures, their corresponding dimensions are in a particular ratio. This ratio is called the scale ratio. Also, for similar figures, the corresponding angles are equal. 


Consider two similar triangles. They would be mathematically represented as:


∆PQR ~ ∆ SUV


Since they are similar, the following is true for them.

  • PQ/SU = QR/UV = PR/SV = scale ratio

  • ∠P = ∠S, ∠Q = ∠U, ∠R = ∠V (corresponding angles)

For congruent figures, their scale ratio is equal to one, because their dimensions are equal.


Area of Similar Figures

The scale ratio is the ratios in which the dimensions of the two similar figures exist. The area is written in square units. The ratio of the areas of similar figures exists in the square of the ratio of the side. This can be mathematically represented as follows, where SFA is the scale factor for the area of a similar figure.


SFA = SF2


The Volume of Similar Figures

Volume is written in cube units. The ratio of the volume of similar figures exists in the cube of the ratio of the side. This can be mathematically represented as follows, where SFv is the scale factor for the volume of a similar figure.


SFV = SF3


Solved Examples


Example 1: Prove that the ratio of the area of two similar figures is the square of the ratios of their sides using the below figures.


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Solution:

Ratio of Sides = HE/DA = EF/AB


= 4/2 = 8/4


= 2


Square of Ratio of Sides = 22 = 4


Area of Rectangle = l x b


Area (ABCD) = 4 x 2


= 8 cm2


Area (EFGH) = 8 x 4


= 32 cm2


Ratio of Areas = Area (EFGH) / Area (ABCD)


= 32 / 8


= 4


Ratio of Areas = Square of Ratio of Sides = 4


Hence, proved.


Example 2: The below two rectangles are similar. Find the ratio of their perimeters. Establish a relationship between the ratio of sides to the ratio of perimeters. 


Solution:


Ratio of Sides = HE/DA = EF/AB


= a/4 = 12/6  = 2


a = 4 x 2 = 8 cm 


Perimeter of Rectangle = 2 (l + b)


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Perimeter (ABCD) = 2 (6 + 4)


= 2 (10)


= 20 cm


Perimeter (EFGH) = 2 (12 + 8)


= 2 (20) 


= 40 cm


Ratio of Perimeters = Perimeter (EFGH) / Perimeter (ABCD)


= 40 / 20


= 2


Ratio of Perimeters = Ratio of Sides = 2


This is the relationship between the ratio of perimeters to ratio of sides. 


Example 3: The perimeters of two similar triangles is in the ratio 2 : 3. The sum of their areas is 169 cm2. Find the area of each triangle.


Solution:


We know that the perimeters of two similar triangles are in the ratio 2 : 3


Then,

The perimeter of the 1st  triangle =  2x

The perimeter of the 2nd  triangle = 3x


We know that the ratio of the areas is equal to the square of ratios of the sides, and the ratios of the sides are equal to the ratio of the perimeters. Hence, the ratio of the areas is equal to the square of ratios of the perimeters.


Area of 1st triangle: Area 2nd  triangle= (2x)2 : (3x)2


Area of 1st triangle: Area 2nd triangle=  4x2: 9x2



The sum of the areas is 169 cm2


Then, 4x2 + 9x=  169


13x2 = 169


x2 = 13


Therefore, the area of the triangles are:


Area of 1st  triangle =  4(13) =  52 cm2

Area of 2nd triangle =  9(13) =  117 cm2

FAQ (Frequently Asked Questions)

1. What is the Difference Between Similarity and Congruence?

Similar objects are the same in shape, whereas congruent objects are the same in shape and size. Areas of two similar figures can be different. The areas of two congruent figures are always equivalent because their dimensions are equal. The ratios of the corresponding sides of two similar figures are equivalent. The ratio of the corresponding sides of the two congruent figures is always one. We can say that all congruent figures are similar, but only some similar figures are congruent.