Pythagorean theorem also known as Pythagoras’ theorem can be defined as a relation among the three sides (hypotenuse, base, perpendicular) of a right-angled triangle. It states that the sum of squares of two small sides(base and perpendicular) is equal to the square of the longest side (hypotenuse).

This theorem is named after the Greek philosopher Pythagoras who was born around 570 BC.

Pythagorean Theorem Formula

a² + b²  = c²

Here c is denoting the length of the hypotenuse and a and b are denoting the lengths of the perpendicular and the base.

Therefore,

Hypotenuse² = Perpendicular² + Base²

Example: Take a right-angled triangle whose sides are 3, 4, 5, now prove the Pythagorean theorem.

Solution:

Sum of squares of two small sides = 3² + 4² = 25

Square of the longest side = 5² = 25

Hence we can see that

sum of squares of two small sides = square of the longest side        

                                                25 = 25

Pythagorean theorem shows the relation in the sides of a right-angled triangle, so if the length of any side is missing, it can be calculated using the Pythagorean Theorem.

If the lengths of both a (Perpendicular) and b (Base) are given, then the length of c can be calculated by using the formula:

c = √(a² + b²)

Similarly a and b can also be calculated if those are missing.

Sample Questions

Question 1: Find the hypotenuse of a right angled triangle whose base is 6 cm and whose height is 8 cm?

Answer: 

Using Pythagorean theorem, a² + b² = c²

So 6² + 8² = c² 

hence c = √(36 + 64)

          c = √100

          c = 10 cm

Question 2: Find whether the given triangle is a right-angled triangle or not, sides are 6, 8, 12?

Answer: 

A right-angled triangle follows the Pythagorean theorem so let’s check it.

Sum of squares of two small sides should be equal to the square of the longest side

So 6² + 8² must be equal to 12²

but 36 + 64 =100 while 12² = 144

Hence it is not a right angled triangle as it is not satisfying the Pythagorean theorem.

Question 3: Find the base of a right angled triangle whose hypotenuse is 13 cm and whose height is 12 cm?

Answer: 

Using Pythagorean theorem, a² + b² = c²

a(perpendicular)= 12, c(hypotenuse) = 13, find b(base)

So b = √(c² – a²)

hence b = √(169 – 144)

           b = √25

           b = 5 cm

Question 4: Find the perpendicular of a right angled triangle whose hypotenuse is 25 cm and whose base is 7 cm?

Answer:

Using Pythagorean theorem, a² + b² = c²

b(base)= 7, c(hypotenuse) = 25, find a(perpendicular)

so a = √(c² – b²)

hence a = √(625 – 49)

          a = √576

          a = 24 cm

Question 5: Find whether the given triangle is a right-angled triangle or not, sides are 10, 24, 26?

Answer: 

A right-angled triangle follows the Pythagorean theorem so let’s check it.

Sum of squares of two small sides should be equal to the square of the longest side

so 10² + 24² must be equal to 26²

100 + 576 = 676 which is equal to 26² = 676

Hence the given triangle is a right-angled triangle because it is satisfying the Pythagorean theorem.