Pythagoras Theorem

Pythagoras theorem is a theorem that states that when a triangle is placed at 90 degrees or is a right-angled triangle, then the square of the 2 sides of the triangle will be similar to the square of the hypotenuse. It is important as Pythagoras’ theorem helps in explaining the relationship between all of the sides of the triangle which is placed at 90 degrees. From this formula, we can easily find out the length of the side which is not known and the angle of the triangle. We can get 3 types of formula by this theorem, one of the hypotenuses, one of the base, and one of perpendicular. These 3 are referred to as the sides of the triangle. 

PYTHAGORAS THEOREM FORMULA:

There is a right-angle triangle with 3 sides named base, perpendicular, and hypotenuse. Let’s assume a triangle to be ABC, then

A will be the perpendicular, B will be the base C will be the hypotenuse

As the definition of the Pythagoras theorem states that the square of hypotenuse should be equal to the square of the base and perpendicular which are the other sides of the triangle so, 

The formula of Pythagoras theorem is:

Hypotenuse2 = Perpendicular2 + Base2. 

PYTHAGORAS THEOREM-PROOF:

A right-angled triangle XYZ is given which is right-angled at the point Y. 

Here to prove : XZ2 = XY2 + YZ2. 

We need to construct a perpendicular YD that meets XZ at D. 

Proof:

As we already know that ∆ XDY = ∆ XYZ

So, 

XDXY = XYXZ

Or we can also say that, 

XY2 = XD * XZ…………………………………………………………….equation 1 

We also know that 

∆YDZ = ∆ XYZ 

So, 

ZDYZ = YZXZ

Or we can also say that, 

YZ2 = ZD * XZ …………………………………………………………..equation 2 

On addition of both the equations, the result is:

XY2 + YZ2 = XD * XZ + ZD * XZ

XY2 + YZ2 = XZ (XD + ZD)

As XD + ZD = XZ 

So, XZ2 = XY2 + YZ2. (Hence proved)

APPLICATIONS OF PYTHAGORAS THEOREM:

Applications of Pythagoras theorem are:

1. It is used for finding out the lengths of all the 3 sides of the triangle. (The triangle should be at 90 degrees).

2. Pythagoras’ theorem helps in recognizing faces in cameras that are used for security purposes. 

3. Pythagoras theorem is used by architects as it helps them in the field of construction and the field of engineering. 

4. It also helps in the field of trigonometry as it helps in finding out many ratios which are there in trigonometry like sin, cos, tan, cosec, etc. 

5. It also helps in finding out the shortest route that can be used for going to a particular place which will save a lot of time. This also helps in navigation. 

6. Pythagoras’ theorem helps in getting the formula of the base, perpendicular, and the hypotenuse, which are the 3 sides of the right-angled triangle. 

PYTHAGORAS THEOREM EXAMPLES:

1. The length of the 2 sides of the triangle is 4 and 3cm. Find out the hypotenuse of the right-angled triangle. 

Answer:

The base and perpendicular of the triangle are 4cm and 3cm respectively. We have to find out the hypotenuse. 

Hypotenuse2 = Base2 + Perpendicular2

Hypotenuse2 = 42 + 32

Hypotenuse2 = 16+9

Hypotenuse = 5cm. 

So the hypotenuse of the right-angled triangle is 5cm. 

2. The hypotenuse of the right angle triangle is given as 20 inches and one of the sides of that triangle is given as 16 inches. So what will be the other side of the third side of the triangle? 

Answer:

Here, the base of the right angle triangle is given 16 inches and the hypotenuse is given as 20 inches and we have to find out the perpendicular. 

So according to the formula,

Hypotenuse2 = Base2 + Perpendicular2

Perpendicular2 = (20)2 – (16)2

Perpendicular = 144

So, the perpendicular of the right angle triangle = 12 inches.