You are familiar with triangles and many of their properties from your earlier classes. In Class IX, you have studied congruence of triangles in detail. Recall that two figures are said to be congruent, if they have the same shape and the same size. In this chapter, we shall study about those figures which have the same shape but not necessarily the same size. Two figures having the same shape (and not necessarily the same size) are called similar figures. In particular, we shall discuss the similarity of triangles and apply this knowledge in giving a simple proof of Pythagoras Theorem learnt earlier.
In Class IX, you have seen 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.
Note that some are congruent and some are not, but all of them have the same shape. So they all are, what we call, similar. Two similar figures have the same shape but not necessarily the same size.
What can you say about the similarity of two triangles?
You may recall that triangle is also a polygon. So, we can state the same conditions for the similarity of two triangles.
That is: Two triangles are similiar, if
(1) their corresponding angles are equal and (2) their corresponding sides are in the same ratio (or proportion).
In the previous section, we stated that two triangles are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion)
You have learnt that in two similar triangles, the ratio of their corresponding sides is
the same. Do you think there is any relationship between the ratio of their areas and
the ratio of the corresponding sides?
You know that area is measured in square units. So, you may expect that this ratio is the square of the ratio of their corresponding sides. This is indeed true and we shall prove it in the next theorem.
You are already familiar with the Pythagoras Theorem from your earlier classes. You
had verified this theorem through some activities and made use of it in solving certain
problems. You have also seen a proof of this theorem in Class IX. Now, we shall prove
this theorem using the concept of similarity of triangles. In proving this, we shall make use of a result related to similarity of two triangles formed by the perpendicular to the hypotenuse from the opposite vertex of the right triangle.
Now, let us take a right triangle ABC, right angled at B. Let BD be the perpendicular to the hypotenuse AC
In this chapter you have studied the following points :
1. Two figures having the same shape but not necessarily the same size are called similar figures.
2. All the congruent figures are similar but the converse is not true.
3. Two polygons of the same number of sides are similar, if
(1) their corresponding angles are equal and
(2) their corresponding sides are in the same ratio (i.e., proportion).
4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.
5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.