Pythagorean Theorem in Grade IX

Hello friends, in today's session we are going to learn about Pythagorean Theorem. The theorem gives a simple relation between the three sides of a right angle triangle.

According to the theorem, it 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: a² + b² =c², 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 is also used to find whether a triangle is acute, obtuse or right angle.

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

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

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

Pythagorean triplet is a set of three positive integers a, b and 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 of the triplets below 100.

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

1. if two sides of the triangles are 5 and 12, then find the length of the hypotenuse?

By using Pythagoras theorem. a ² + b ² = c ². in here a = 5 and b = 12, so the value of c will be c ² = 25 + 144.

c ² = 169

c = 13.

2. if one side of the triangle is 9 and the hypotenuse is 15, then find the other side.

In this problem a = 9 and c= 15, then the third side will be b.

b ² = c ² - a ^2.

b ² = 225-81

b ² = 144

b = 12.

3. check whether the triangle with given sides is a right triangle or not. a = 10, b = 24 and c = 26.

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

26² = 10² + 24^2.

676 = 100 + 576.

676 = 676,

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