Similar Figures: Definition, Properties, and Examples

Similar Figures: We know that all circles with the same radii are congruent, all squares with the same side lengths are congruent, and all equilateral triangles with the same side lengths are congruent. Now, consider any two circles. Since all of them do not have the same radius, they are not congruent. But all of them have the same shape. So, they are all called similar.

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Congruent Figures

Geometrical figures are classified based on their shapes using the concept of congruence. If two geometrical figures have the same shape and size, they are said to be congruent. Using the method of superposition, two line segments and two angles can be found to be congruent. The method can be applied to any two figures of the same type.

Two line segments are congruent if and only if their lengths are equal. In other words, if two line segments \(AB\) and \(CD\) are congruent if and only if \(AB=CD\).

Two angles are congruent if and only if their measures are equal. Two angles \(B A C\) and \(E D F\) are congruent if \(\angle B A C=\angle E D F\).

Similar Figures

Can a circle and a square be similar? Can a triangle and a square be similar? These questions can be answered by just looking at the figures below:

By observation, we can tell that they are not similar. Now, can we say that two quadrilaterals \(A B C D\) and \(P Q R S\) are similar?

These figures appear to be similar, but we cannot be certain about them. Therefore, we must define the similarity of figures and, based on this definition, some rules to decide whether the two given figures are similar.

Geometric figures that have the same shape but different sizes are known as similar figures.

Any two line segments are always similar, but they need not be congruent. They are congruent if their lengths are equal.

Any two circles are similar but not necessarily congruent. They are congruent if the radii are equal.

Any two squares and any two equilateral triangles are similar.

What can you say about the two photographs of the same size of the same person, one at the age of \(10\) years and the other at \(40\) years? Are these photographs similar? These photographs are of the same size, but certainly, they are not of the same shape. So, they are not similar.

What does the photographer do when he prints photographs of different sizes from the same negative? You must have heard about the stamp size, passport size, and postcard size photographs. He takes a photograph on a small size film, says \(35 \mathrm{~mm}\), and then enlarges it into a bigger size, say \(45 \mathrm{~mm}\) (or \(55 \mathrm{~mm})\). Thus, if we consider any line segment in the smaller photograph (figure), its corresponding line segment in the bigger photograph will be \(\frac{45}{35}\) (or \(\frac{55}{35}\)) of that of the line segment.

This means that every line segment of the smaller photograph is enlarged (increased) in the ratio \(35: 45\) (or \(35: 55\)). But, on the other hand, it can also be said that every line segment of the bigger photograph is reduced (decreased) in the ratio \(45: 35\) (or \(55: 35\)). Further, suppose you consider inclinations (or angles) between any pair of corresponding line segments in the two photographs of different sizes. In that case, you shall see that these inclinations(or angles) are always equal. This is the nature of the similarity of two figures and, in particular, of two polygons.

So, two polygons of the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio (or proportion).

Similar Polygons

Two polygons having the same number of sides are said to be similar. If:

1. Their corresponding angles are equal, and
2. The length of their corresponding sides are proportional

If two polygons \(A B C D E\) and \(P Q R S T\) are similar, we write, \(ABCDE \sim PQRST,\) where the symbol stands for ‘is similar to.’

From the above definition, if two polygons \(A B C D E\) and \(P Q R S T\) are similar, then it follows that:

Angle at \(A=\) Angle at \(P\), Angle at \(B=\) Angle at \(Q\), Angle at \(C=\) Angle at \(R\), Angle at \(D=\) Angle at \(S\), Angle at \(E=\) Angle at \(T\).

So, \(\frac{A B}{P Q}=\frac{B C}{Q R}=\frac{C D}{R S}=\frac{D E}{S T}=\frac{E A}{T P}\)

For the similarity of polygons with more than three sides, the two conditions given in the definition are not independent of each other, i.e., either of the two conditions without the other is not sufficient for polygons with more than three sides to be similar. In other words, if the corresponding angles of two polygons are equal but lengths of their corresponding sides are not proportional, the polygons need not be similar. Similarly, if the corresponding angles of two polygons are not equal but the length of the corresponding sides are proportional, then the polygons need not be similar. 

Triangles are a special type of polygons. In triangles, If either of the two conditions given in the above definition holds, then the other holds automatically.

Properties of Similar Polygons

1. The corresponding angles are equal or congruent. This means both interior and exterior angles are the same.
2. The ratio of the corresponding sides are proportional.

Similar Triangles

Two triangles are said to be similar if their corresponding angles are equal and corresponding sides are proportional.

If two triangles \(\triangle A B C\) and \(\triangle D E F\) are similar, then \(\angle A=\angle D, \angle B=\angle E, \angle C=\angle F\) and \(\frac{A B}{D E}=\frac{B C}{E F}=\frac{A C}{D F}\).

The two conditions given in the above definition are independent. If either of the two conditions holds, then the other holds automatically. So any one of the two conditions can be used to define similar triangles. If corresponding angles of two triangles are equal, then they are known as equiangular triangles.

Solved Examples – Similar Figures

Q.1. Give two examples of pairs of (i) similar figures (ii) non-similar figures.
Ans: Two different examples of pair of similar figures are:

(a) Any two rectangles

(b) Any two squares

Two different examples of pair of non-similar figures are:

(a) Circle and a triangle

(b) Square and a rectangle

Q.2. State whether the following quadrilaterals are similar or not.

Ans: From the given two figures,
\(\angle S P Q\) is not equal to \(\angle D A B\)
\(\angle P Q R\) is not equal to \(\angle A B C\)
\(\angle Q R S\) is not equal to \(\angle B C D\)
\(\angle R S P\) is not equal to \(\angle C D A\)
Hence, the quadrilaterals are not similar.

Q.3. State whether the following quadrilaterals are similar or not.

Ans: In hexagons \(A B C D E F\) and \(P Q R S T U\)
\(\angle A=\angle P, \angle B=\angle Q, \angle C=\angle R, \angle D=\angle S, \angle E=\angle T, \angle F=\angle U\)
\(\frac{A B}{P Q}=\frac{B C}{Q R}=\frac{C D}{R S}=\frac{D E}{S T}=\frac{E F}{T U}=\frac{F A}{U P}\)
Hence, the two hexagons are similar.

Q.4. In the below figures, \(A B C D\) is a square, and \(P Q R S\) is a rhombus. Check whether they are similar.

Ans: We know that each angle of a square is equal to \(90^{\circ}\).
So, \(\angle A B C\) is not equal to \(\angle P Q R\)
\(\angle B C D\) is not equal to \(\angle Q R S\)
\(\angle C D A\) is not equal to \(\angle R S P\)
\(\angle D A C\) is not equal to \(\angle S P Q\)
So, the given figures are not similar.

Q.5. In the given figure, verify whether quadrilateral \(A B C D\) and \(P Q R S\) are similar or not.

Ans: In the given quadrilaterals \(A B C D\) and \(P Q R S\)
\(\angle A=\angle P=105^{\circ}\)
\(\angle B=\angle Q=100^{\circ}\)
\(\angle C=\angle R=70^{\circ}\)
\(\angle D=\angle S=85^{\circ}\)
Also, \(\frac{A D}{P S}=\frac{D C}{S R}=\frac{C B}{R Q}=\frac{B A}{Q P}=\frac{1}{2}\)
So, the given quadrilaterals \(A B C D\) and \(P Q R S\) are similar.

Summary

By reading the above article, we learned what congruent figures are, what similar figures are, and how to solve similar polygon and similar triangle problems.

Frequently Asked Questions (FAQ) – Similar Figures

Let’s look at some of the commonly asked questions about congruent figures:

Q.1. How do you identify similar figures?
Ans: Geometric figures which have the same shape but different sizes are known as similar figures. So, if two figures have the same shape, then we say both figures are similar.

Q.2. What are the two characteristics of similar figures?
Ans: The two characteristics of similar figures are:
1. Their corresponding angles are equal
2. The length of their corresponding sides are proportional.

Q.3. Are all regular hexagons similar?
Ans: A regular hexagon is one with all equal sides. Since it is made of \(6\) equilateral triangles, all regular hexagons would be similar with equal angles but different sides.

Q.4. How can you tell if the two figures are similar?
Ans: If two figures are similar, then they will have the same shape but different sizes.

Q.5. What is a real-life example of a similar figure?
Ans: Two square tiles, the rare wheels of a car, LED TVs of different inch sizes, etc., can be real-life examples of similar figures.

Q.6. What are congruent figures?
Ans: The concept of congruence is used to classify the geometrical figures based on their shapes. Two geometrical figures are said to be congruent if they have exactly the same shape and same size.

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