What Is Pythagoras Theorem?

Pythagoras Theorem Statement

Pythagoras theorem asserts that In a right-angled triangle, the square of the hypotenuse side is equivalent to the total of squares of the other two sides . The sides of this triangle have been defined as Perpendicular, Base and Hypotenuse. Here, the Hypotenuse is the longest side since it is against the angle of 90 . The sides of a right triangle (assume a, b and c), which have positive integer values while squared, are put into an equalization, also called a Pythagorean triple.

History

The theory is named after a Greek Mathematician named Pythagoras.

Pythagoras Theorem Formula

Suppose there is a triangle with

a - perpendicular,

b - the base,

c - Hypotenuse.

Hypotenuse² = Perpendicular² + Base²

c² = a² + b²

The side contrary to the right angle (90 ) is the longest side (also known as Hypotenuse) since the side contrary to the greatest angle is the largest.

Consider 3 squares of sides a, b, c positioned on the three sides of a triangle holding the same sides as presented.

With Pythagoras Theorem

square A (Area)+ square B (Area) = square C. (Area)

Example

The cases of theorem and based on the observation given for right triangles is given below:

Consider a right triangle, given below:

Find the value of x.

Since X is the side opposite to right angle, it is a hypotenuse.

Now, by the theorem we know

Hypotenuse² = Base²+ Perpendicular²

x²= 8²+ 6²

x²= 64+36 = 100

x = 100 = 10

Therefore, the value of x is 10.

Proof Pythagoras Theorem

Given:

A right-angled triangle ABC, right-angled at B.

Prove: AC² = AB² + BC²

Construction: Form a perpendicular BD joining AC at D.

Proof:

We identify ADB ~ ABC

Therefore, ADAB=ABAC

(corresponding surfaces of similar triangles)

Or, AB² = AD AC .. ..(1)

Also, BDC ~ ABC

Therefore, CDBC=BCAC

(corresponding sides of similar triangles)

Or, BC²= CD AC ..(2)

Adding the equations (1) and (2)

we get,

AB² + BC² = AD AC + CD AC

AB² + BC² = AC (AD + CD)

Since, AD + CD = AC

Therefore, AC²= AB² + BC²

Hence, the Pythagorean theorem is proved.

Note: Pythagorean theorem is only applicable to the Right-Angled triangle.

Applications of Pythagoras Theorem

To check if the triangle is a right-angled triangle or not. For every right-angled triangle, we can find the length of any side if the other two sides are given. To find the diagonal of a square.

Useful For

Pythagoras theorem is helpful to determine the sides of a right-angled triangle. Suppose if we know the two sides of a right triangle, then we can find the third side.

How to use it?

To use this theorem, recollect the formula given below:

c² = a²+ b²

(Let the right triangle sides be a, b and c)

For instance, if the value of a = 6 cm, b = 8 cm, then find the value of c.

We know,

c²= a² + b²

c² =6²+8²

c² = 36 + 64

c² = 100

c = 10

Hence, the 3rd side is 5 cm.

As we can see,

a + b > c

6 +8 > 10

14 > 10

Hence the hypotenuse is c = 10 cm

How to check whether a triangle is a right-angled triangle?

Let us explain this statement with the help of an example.

Assume a triangle with sides 20, 48, and 52 are given.

52 is the longest side.

It also satisfies the condition, 20 + 48 > 52

We know,

c² = a² + b² (1)

So, let a = 20, b = 48 and c = 52

First we will find R.H.S. of equation 1.

a² + b²= 20² + 48² = 400 + 2304 = 2704

Now, taking L.H.S, we get

c²= 52² = 2704

We can see,

LHS = RHS

Hence, the above triangle is a right triangle

Pythagorean Theorem Problems

Ques 1: The sides of a triangle are 15, 36 39 units.

Solution: From Pythagoras Theorem, we have

Perpendicular² + Base² = Hypotenuse²

Let,

Perpendicular = 36 units

Base = 15 units

Hypotenuse = 39 units {since it is the longest side measure}

36² + 15² = 39²

1296+ 225 = 1521

1521 = 1521

L.H.S. = R.H.S.

Therefore, the angle opposite to the 39 units side will be a right angle.

Problem 2: The 2 sides of a right-angled triangle are given. Find the third side.

Solution: GivenPerpendicular = 12 cm

Base = b cm

Hypotenuse = 13 cm

As per the Pythagorean Theorem, we have

Perpendicular² + Base²= Hypotenuse²

12²+ b² =13²

225 + b²= 289

b² = 289 225

b²= 64

b = 64

Therefore, b = 8 cm

Problem 3: Assume the side of a square to be 3 cm. Find the length of the diagonal.

Solution- Given

Sides of a square = 3 cm

To get- The length of diagonal ac.

Suppose triangle abc (or can also be acd)

² +(bc) ² = (ac) ²

² +(3) ²= (ac) ²

²

²

c = (a² + b²)

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