We all are familiar with triangles and many of their properties, we have learnt in previous classes.
We have learnt that two figures are said to be congruent, if they have the same shape and the same size.
To finding all the heights and distances have been found out using the idea of indirect measurements, which is based on the principle of similarity of figures. We will learn here to finding the solutions.
We have learnt and 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.
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).
The same ratio of the corresponding sides is referred to as the scale factor (or the Representative Fraction) for the polygons.
Two polygons of the same number of sides are similar, if
All the corresponding angles are equal and
All the corresponding sides are in the same ratio (or proportion).
Either of the above two conditions (i) and (ii) of similarity of two polygons is not sufficient for them to be similar.
If corresponding angles of two triangles are equal, then they are known as equiangular triangles.
That is:
Two triangles are similiar, if
their corresponding angles are equal and
their corresponding sides are in the same ratio (or proportion).
The ratio of any two corresponding sides in two equiangular triangles is always the same.
If corresponding angles of two triangles are equal, then they are known as equiangular triangles. A famous Greek mathematician Thales gave an important truth relating to two equiangular triangles which is as follows:
The ratio of any two corresponding sides in two equiangular triangles is always the same.
It is believed that he had used a result called the Basic Proportionality Theorem (now known as the Thales Theorem) for the same.
Theorem 6.1 : If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Theorem 6.2 : If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.
We learnt that two triangles are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (or proportion). Here we will see that A corresponds to D, B corresponds to E and C corresponds to F. Symbolically, we write the similarity of these two triangles as ‘△ABC ~ △DEF’ and read it as ‘triangle ABC is similar to triangle DEF’. The symbol ‘~’ stands for ‘is similar to’. Recall that you have used the symbol ‘≅’.
Theorem 6.3 : If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio (or proportion) and hence the two triangles are similar.
Theorem 6.4 : If in two triangles, sides of one triangle are proportional to (i.e., in the same ratio of ) the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similiar.
Theorem 6.5 : If one angle of a triangle is equal to one angle of the other triangle and the sides including these angles are proportional, then the two triangles are similar.
As we have learnt that in two similar triangles, the ratio of their corresponding sides is the same.
Here we will learn there is any relationship between the ratio of their areas and the ratio of the corresponding sides.
Theorem 6.6 : The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
We have already familiar with the Pythagoras Theorem from your earlier classes. Here we will 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.
△ ADB ≅ △ ABC
and △ BDC ≅ △ ABC
So, △ ADB ≅ △ BDC
Theorem 6.7 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to
the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.
Theorem 6.8 : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth).
Theorem 6.9 : In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.