How to Identify and Solve Problems on 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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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.
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.
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.
FAQs on Similar Figures in Maths: Concepts, Formulas & Stepwise Solutions
1. What is the definition of similar figures in Class 10 Maths?
In geometry, two figures are defined as similar if they have the exact same shape, but their sizes may be different. For any two polygons to be similar, they must satisfy two key conditions: all corresponding angles must be equal, and the ratio of the lengths of all corresponding sides must be constant. This constant ratio is known as the scale factor or ratio of similarity.
2. How are similar figures different from congruent figures?
The primary difference lies in their size. Congruent figures are identical in both shape and size, meaning the ratio of their corresponding sides is exactly 1. Similar figures, however, only need to have the same shape, allowing for different sizes. Essentially, all congruent figures are similar, but not all similar figures are congruent. For example, two circles are always similar, but they are only congruent if their radii are equal.
3. What are the conditions required to prove two triangles are similar?
According to the CBSE syllabus, you can prove two triangles are similar using any one of the following three criteria:
- Angle-Angle (AA) Similarity: If two angles of one triangle are equal to two corresponding angles of another triangle, then the two triangles are similar.
- Side-Side-Side (SSS) Similarity: If the corresponding sides of two triangles are in the same ratio (proportional), then the triangles are similar.
- Side-Angle-Side (SAS) Similarity: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.
4. What are some real-life examples of the application of similar figures?
The concept of similar figures is widely used in real life. Common examples include creating architectural blueprints and maps, where a large structure or area is scaled down proportionally. It's also used in photography for enlarging or reducing images, in engineering for creating scale models of cars or buildings, and even in basic trigonometry to calculate the height of tall objects like trees or buildings by comparing the shadow they cast to the shadow of a smaller, known object.
5. Why is the concept of a 'scale factor' important for similar figures?
The scale factor is a crucial concept because it acts as the bridge between two similar figures. It is the constant ratio of corresponding side lengths. This single number tells us exactly how much larger or smaller one figure is compared to the other. In practical applications, the scale factor is essential for accurately resizing objects, such as scaling up a design from a blueprint or calculating unknown distances on a map.
6. If two triangles are similar, what is the relationship between their areas?
This is a key theorem in the study of similar figures. If two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides. For example, if the ratio of the sides of two similar triangles is 2:3, then the ratio of their areas will be (2/3)², which is 4:9. This principle is very important for solving problems involving the areas of similar geometric shapes.
7. Are all circles and all squares always similar to each other?
Yes. All circles are inherently similar to each other because they all share the same perfect circular shape, differing only by their radius. Likewise, all squares are similar to each other. This is because every square has four 90-degree angles, satisfying the equal-angle condition, and the ratio of any two corresponding sides will always be constant. This is not true for all rectangles or all triangles, which require specific angle and side ratio conditions to be met.
8. Can a right-angled triangle be similar to an equilateral triangle?
No, a right-angled triangle can never be similar to an equilateral triangle. For two triangles to be similar, all their corresponding angles must be equal. An equilateral triangle has all angles equal to 60 degrees. A right-angled triangle must have one angle of 90 degrees. Since the angles can never match, these two types of triangles can never be similar.
