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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In-Depth Concept of Similar Figure
In general, the opinion to say that something is similar to something else is to mention that the two things share common characteristics. For example, you and you’re friend might be brainstorming on how to solve a particular problem, and you tell him the approach you would take. He then will tell you that your approach is similar to the plan he was thinking of. This means that your ideas on how to solve the problem are much the same but could have small differences as well.
In mathematics, saying that two figures are similar means that they share a common shape. They can be different sizes, but they must have the same shape.
Definition of Similar Figure
Two figures are explained to be similar if they are the same shape. In more mathematical language, two figures are similar if their corresponding angles are harmonious, and the ratios of the lengths of their equivalent sides are equal.
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:
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∆PQR ~ ∆ SUV
Since they are similar, the following is true for them.
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PQ/SU = QR/UV = PR/SV = scale ratio
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∠P = ∠S, ∠Q = ∠U, ∠R = ∠V (corresponding angles)
For congruent figures, their scale ratio is equal to one, because their dimensions are equal.
Similar Figures Area and Volume
If two figures are similar, then their corresponding sides are proportional or it can also be explained as when the ratio of their sides is equal.
For example; if we take the ratio of their surface areas, then it will be identical to the square of the ratio of sides. The ratio of the volume of two similar figures will be equal to the cube of the ratio of the length of sides.
Hence, based on the declarations mentioned above, the scale factors of area and volume can be represented as;
SFA = SF2
SFV = SF3
where SFA is the scale factor of surface area and SFV is the scale factor of volume
Example of Similar Figures:
Following are the examples of similar figures:
1. Pair of equilateral triangles
2. Pair of squares
3. Pair of circles
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 cm
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
= 4cm2
Ratio of Areas = Square of Ratio of Sides = 4cm2
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
= 2cm
Ratio of Perimeters = Ratio of Sides = 2cm
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 + 9x2 = 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
Conclusion
This is all about the concept of similar figures, its explanation with solved examples. Understand how similar figures are explained here and develop your concepts well. Focus on how the concept is used to solve problems.