Pythagorean theorem is one of the most used and important topics of mathematics. It is based upon the relationship between hypotenuse, base, and perpendicular of a right-angle triangle. If a triangle has a 90-degree angle or a right angle it is called a right-angle triangle. The opposite side of the right angle is called the hypotenuse and the other two sides are called base and perpendicular respectively. The hypotenuse is the longest side of a right-angled triangle. According to Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the square of the base and perpendicular side. So,

The origin of the Pythagorean theorem: Pythagorean theorem is first introduced by the famous Greek mathematician and philosopher Pythagoras of Samos. He lived a life of a monk and he started with a group of mathematicians. His mathematical theories have a great impact on present days mathematical calculations. The name of the Pythagorean theism is named after him as he was introduced. This theorem has been solved different kinds of problems regarding triangles. Though the theorem was introduced long years ago, this theorem has a deep impact on present days calculation also. This theory helps to differentiate the right-angle triangle from other types of triangles. This theorem is also applied to solve and calculate many real-life problems also.

What is the Pythagorean Theorem?

The Pythagorean theorem states that in a right-angled or 90-degree angled triangle, the square of the hypotenuse is equal to the sum of the square of the base and perpendicular side of it. So,

If we observe a right-angled triangle and mark its angles as ABC, according to the Pythagorean theorem, if the hypotenuse is BC, AB is the base and AC is the perpendicular or the altitude of the triangles, then

Pythagorean Theorem formula:

Pythagorean theorem helps solve any kind of right-angled triangle problem. The Pythagorean equation is

(If the hypotenuse is marked as ‘c’ and the other two legs are ‘a’ and ‘b’)

 This formula can be applied to any right-angled triangle i.e., a triangle having a 90-degree angle.

Now we will take an example. If there is a right-angle triangle in which the hypotenuse(c) is 5cm. and the altitude(a) and the base(b) is 4cm. and 3cm. respectively. Then according to the Pythagorean theorem

So, in this case,

Therefore, 25 = 16 + 9

25 = 25

Application of Pythagorean Theorem Formula and some calculations: Now we will discuss the application of the Pythagorean theorem and will give some examples which will help you to master the calculations easily. The theorem is frequently used in trigonometric calculation also. We can apply cos, sine, tan to find out the length of the sides of a right-angled triangle.

 The formula of the Pythagorean theorem is generally used to calculate the sides of a right-angle triangle. If the length of two sides of a right-angled triangle is given, we can calculate the length of another side from this theorem.

Have examples.

If the length of a and b are 12 and 5. Calculate the length of c.

As

Therefore

144 + 25 =

169 =

So, c is √169 =13

From this theorem, we can find out the hypotenuse. In the above example, the value of c is hypotenuse as it is the longest among all sides.

The Pythagorean theorem helps to know that if a triangle is a right-angle triangle or not. When the three sides of a triangle are given and we need to identify whether the triangle is a right-angle triangle or not we have to calculate it through this formula. To clarify this statement, let’s see the example:

The lengths of the three sides of a triangle are 10,24, and 26. We have to find out whether it is a right-angled triangle or not?

Here 26 is the length of the hypotenuse because it is the longest side. So, 26 is c and suppose 10 is a and 24 is b.

As we know,

If it is a right-angle then,

100 + 576 = 676

676 = 676

Therefore, it is proved that it is a right-angled triangle.

So, this Pythagorean theorem is very important in Mathematical calculation. You can install the Cuemath app and you will find numerous examples of the application of the Pythagorean theorem and you can learn and improve your math knowledge with the help ofCuemath.