# Similar Figures

Similar figures are two figures having the same shape. The objects which are of exactly same shape and size are known as congruent objects. For example, both the front wheels of a car, both hands of a person etc. are examples of congruent objects.

When two figures have same shape but their sizes are different, then such objects are called similar figures. For example: Different sized photographs of a person i.e. stamp size, passport size etc. depict the similar objects but are not congruent.

Certain geometrical shapes or figures are always similar in nature. Consider a circle, if the radius of the circle keeps on changing, its shape still remains the same. Therefore, it can be said that all the circles with different radii are similar to each other. The figure given below represents the concentric circles whose radii are different but all of them are similar. Although these circles have the same shape but their sizes are different and therefore these are not congruent. Consider the figures shown below. All of them are similar to each other as they have the same shape but they are not congruent. It has to be kept in mind that whenever we talk about similar figures, we consider their shapes only, irrespective of their sizes. Thus we can conclude that all congruent figures are similar, but all similar figures are not congruent. If we observe all the figures which are similar, we can see that for any n-sided polygon, the inclination angles of each of the line segments are always the same, irrespective of the size of the figure. Thus for any two n-sided polygon of same number of sides, it can be said that they are similar if,

1. Corresponding angles of both the polygons are equal, and
2. Corresponding sides of both the polygons are in same ratio.

It can be said that congruence is a special case of similarity when the ratio of the sides of the figure is 1. In other words, it can be said that when two similar figures become congruent, then their length of corresponding sides becomes equal, as the ratio of their corresponding sides become 1.

The difference between similar and congruent figures must be clear by now. Consider the following polygon and check whether these polygons are similar or not. In the figures given above, as all the angles are equal, first condition, i.e. ‘corresponding angles of both the polygons should be equal’ is easily satisfied. Now, observing the sides of the polygon it can be seen that;

$\frac{PS}{AB}$ = $\frac{PQ}{AD}$ = $\frac{BC}{QR}$ = $\frac{CD}{RS}$ = $\frac{1}{2}$

Therefore we can say that the polygons ABCD and PSRQ are similar to each other.