Converse of Pythagoras’ theorem

Here, we will be talking about the converse of Pythagoras’ Theorem first to have a better understanding of the concept and then will be talking about the converse Pythagoras’ theorem. Here we are already assuming that you all are familiar with the Pythagoras Theorem. But still, to revise, we will just define the Pythagoras theorem in a sentence and then go ahead with vice versa which is the converse of  Pythagoras Theorem. 

Pythagoras Theorem-

Pythagoras theorem talks about the relationship between all three sides of a right-angled triangle. This theorem states that in a right-angled triangle, the square of the hypotenuse is always equal to the sum of the squares of the other two sides. Now let us talk about the converse of this theorem.      

The converse of Pythagoras Theorem-      

According to the converse of  the Pythagoras Theorem “If in a triangle, a square of one side is equal to the sum of squares of other two sides, then the triangle must be a right angle triangle.”

If, H2 = B2 + P2

Then, the triangle is a right-angled triangle.

Proof Of Converse of Pythagoras Theorem-

To Prove: If the measure of three sides of a triangle is a, b and c and a2  = b2  + c2, then the triangle is a Right angle triangle.

Proof: Draw another triangle PQR, such that AC = PQ = b and BC = QR = a.

In △PQR, by Pythagoras Theorem:

PR2 = PQ2 + QR2 = b2 + a2 …………(1)

In △ABC, by Pythagoras Theorem:

AB2 = AC 2 + BC2 = b2 + a2 …………(2)

From equation (1) and (2), we have;

PR2 = AB2

PR = AB

⇒ △ ACB ≅ △PQR (By SSS postulate)

⇒ ∠Q is a right angle

Thus, △PQR is a right triangle.

Hence, we can say that the converse of the Pythagorean theorem holds here.

Hence Proved.

As per converse of Pythagoras theorem, The formula for right angle triangle will be:

a2 + b2 = c2

a, b, c = Sides of triangle

Applications of the converse of Pythagoras Theorem

Generally, the Converse of Pythagoras theorem is used to determine whether a triangle is right-angled or not. If we know about the type of triangle and get to know that the triangle is at the right angle so we have an angle 90 degrees, that will help us a lot in the construction of the triangle.

Solved Examples-

1. Given, the three sides of a triangle are a = 5,b = 12 and c = 13. Check for the given triangle is a right triangle or not?

Sol- Given,

a = 5

b = 12

c = 13

We know that according to the converse of Pythagoras’ theorem

a2 + b2 = c2

Or

c2 = a2 + b2

On Substituting the given values in the above equation,

132 = 52 + 122

169 = 25 + 144

169 = 169

RHS = LHS

That means, the given triangle is a right triangle.

2) Given that the three sides of a triangle are a = 7, b = 11, and c = 13. Check for the given triangle is a right triangle or not?

Sol: Given;

a = 7

b = 11

c = 13

We know that according to the converse of Pythagoras’ theorem

c2 = a2 + b2

On Substituting the given values in the above equation,

132 = 72 + 112

169 = 49 + 121

169 = 170

LHS ≠ RHS

Therefore, it is proved that the given triangle is not a right triangle.

3) Given that the three sides of a triangle are a = 4, b = 6, and c = 8. Check for the given triangle is a right triangle or not?

Sol: Given: a = 4, b = 6, c = 8

We know that according to the converse of Pythagoras’ theorem

c2 = a2 + b2

On Substituting the given values in the above equation,

82 = 42 + 62

64 = 16 + 36

64 = 52

LHS ≠ RHS

Therefore, it is proved from here that the given triangle is not a right triangle.

Conclusion-

We have studied the Pythagoras Theorem as well as the converse of the Pythagoras theorem. We have even talked about its application that we can use to check whether a triangle is right-angled or not. We have also discussed the examples to understand the concept in a better way.