Converse of Pythagoras Theorem
The converse of the Pythagoras theorem states that,
Given a triangle with sides of length a, b, and c, if a2 + b2 = c2, then the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with sides a, b and c as a consequence of the converse of the triangle inequality.
Converse of Pythagoras Theorem Proof
For a triangle with the length of its sides a, b, if c2 = a2 + b2, we need to prove that the triangle is right-angled.
We assume that it satisfies c2 = a2 + b2, and by looking into the diagram, we can tell that ∠C = 90°, but to prove it, we require another triangle △EGF, such as AC = EG = b and BC = FG = a.
In △EGF, by Pythagoras Theorem:
⇒ EF2 = EG2 + FG22 = b2 + a2 ⇢ (1)
In △ABC, by Pythagoras Theorem:
⇒ AB2 = AC2 + BC2 = b2 + a2 ⇢ (2)
From equation (1) and (2), we have;
⇒ EF2 = AB2
⇒ EF = AB
⇒ △ ACB ≅ △EGF (By SSS postulate)
⇒ ∠G is right angle
Thus, △EGF is a right triangle. Hence, we can say that the converse of the Pythagorean theorem also holds.
Pythagoras Theorem | Formula, Proof and Examples
Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The theorem can be expressed as r2 = p2 + q2, where ‘r’ is the hypotenuse and ‘p’ and ‘q’ are the two legs often called perpendicular and base of the triangle.
Pythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem.
In this article, we will learn about the Pythagoras theorem statement, its formula, proof, examples, applications, and converse of Pythagoras theorem in detail.