Pythagoras Theorem in Grade VIII

Hello friends, in today's session we are going to learn about Pythagorean Theorem. This theorem is very useful and gives a very simple relationship between the three sides of a right angle triangle.

The theorem states that :

In any Right Angle Triangle, the area of the square whose Side is the Hypotenuse is equal to the sum of the areas of the square whose sides are the two legs. In algebraic form the theorem can be written as: a ² + b ² = c ², where a and b are the small sides and c is the hypotenuse of the right triangle.

In a right triangle if we know any two sides then we can easily find the other one with the Pythagoras equation.

If we generalize this theorem then we will get the law of cosines, which makes possible the computation of the third side of the triangle.

The converse of the theorem is also true:

For any three positive numbers a,b, and c such that a ² +b ² = c ², there exists a triangle with sides a,b and c, and every such triangle has a right angle between the sides of lengths a and b.

The Pythagoras equation can also be used to check what type of triangle we have, whether it is acute, obtuse or right angled. We can check this by using the following results:-

If a ² + b ² = c ², then the triangle is said to be right angled.

If a ² + b ² > c ², then the triangle is said to be acute angled.

If a ² + b ² < c ², then the triangle is said to be obtuse angled.

Pythagorean triplet is a set of three positive integers a, b and c, they are written in the form ( a, b, c). These triplets satisfy the Pythagoras equation a ² + b ² = c ².

For example - (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37) are some among those which come below 100.

So let's solve some examples based on Pythagoras theorem.

1. if two sides of the triangles are 3 and 4, then find the length of the hypotenuse?

By using Pythagoras theorem. a ² + b ² = c ². in here a = 3 and b = 4, so the value of c will be c ² = 9 + 16.

c ² = 25

c = 5.

2. if one side of the triangle is 9 and the hypotenuse is 41, then find the other side.

In this problem a = 9 and c= 41, then the third side will be b.

b ² = c ² - b ^2.

b ² = 1681-81

b ² = 1600

b = 40.

3. check whether the triangle with given sides is a right triangle or not. a = 11, b = 60 and c = 61.

We can check whether it is a right triangle or not by using the Pythagoras theorem.

61² = 11² + 60^2.

3721 = 121 + 3600.

3721 = 3721,

so these three sides satisfy the Pythagoras theorem, so the triangle formed will be right angled.