# Math՚s: Similarity of Triangles: Some Important Theorems Related to Similarity of Triangles (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Basic Proportionality Theorem

- If a line is drawn parallel to one side of a triangle intersecting the other two sides, the other two sides of the triangle are divided proportionally.
- In the given below, . Thus, according to the proportionality theorem,
- The proportionality theorem can also be conversely stated as- ‘If a line divides any two sides of a triangle in the same ratio, the line is parallel to third side of the triangle’ .
- Thus, if it is given that in , then

.

## Angle Bisector of a Triangle

- The bisector of an interior angle of a triangle divides the opposite side in the ratio of sides containing the angle.
- Thus, according to the above theorem, if is the internal bisector of of , then,

## Some Theorems Related to Right Triangle

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to each other and to the original triangle. Thus, if is right angled at and is drawn perpendicular to the hypotenuse , meeting it in , then as they have the corresponding angles equal.

## Baudhayana or Pythagoras Theorem

In a right triangle, the square of the hypotenuse is equal to sum of the squares of the other two sides. Thus, in a right angled at ,

OR

## The Converse of this Theorem States

In a triangle, if the square on one side is equal to sum of the squares on the other two sides, the angle opposite to first side is a right angle.

1. Find the length of diagonal of a rectangle the lengths of whose sides are 3 cm and 4 cm.

Solution: The adjacent figure, is a rectangle .

Join the diagonal . Now is a right triangle.

i.e.. , the length of diagonal of rectangle is