The Converse of Pythagoras Theorem

What is the Converse of Pythagoras Theorem?

If the square of a side is equal to the sum of the squares of the other two sides then the triangle must be a right angle triangle, this is known to be the Converse of Pythagoras Theorem. This is the Converse of Pythagoras statement. The Pythagorean theorem in Mathematics states that the sum of the square of two sides (legs) is equal to the square of the hypotenuse of a right-angle triangle. But, in the reverse of the Pythagorean theorem, it is known that if this relation satisfies, then the triangle must be a right angle triangle. So, if the sides of a triangle have length, a, b and c and satisfy the given condition c²= a² + b², then the triangle is known to be a right-angle triangle.


What is the Pythagoras Theorem?

Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle θ. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90°.

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The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). The side that is opposite to the angle θ is known as the opposite(O). And the side which lies next to the angle θ is known as the Adjacent(A)

The Pythagoras theorem states that, 

In a right-angle triangle, (Opposite)2+(Adjacent)2= (Hypotenuse)2


Converse of Pythagorean Theorem Proof:

The converse of the Pythagorean Theorem proof is:

Converse of Pythagoras theorem statement: The Converse of Pythagoras theorem statement says that if the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides of a triangle, then the triangle is known to be a right triangle.

That is, in ΔABC if c²= a² + b² then 

∠C is a right triangle, the ΔPQR being the right angle.

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We can prove this by contradiction.

Let us assume that ,

c²= a² + b² in ΔABC and the triangle is not a right triangle.

Now consider another triangle ΔPQR. We construct ΔPQR so that 

PR=a, QR=b and ∠R is a right angle.

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By the Pythagorean Theorem,

(PQ)² = a² + b² 

But we know that , a² + b²  = c² and c=AB.

So, (PQ)² = a² + b² = (AB)² 

That is, (PQ)² = (AB)²

Since the lengths of the sides are PQ and AB, we can take positive square roots.

PQ=AB

That is, all the three sides of the triangle PQR are congruent to the three sides of  the triangle ABC. So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.

This is a contradiction. Therefore, our assumption must be wrong.This is the converse of Pythagoras theorem proof.


Formula of Converse of the Pythagorean Theorem:

As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:

a² + b² = c²

 

Where the variables a, b and c are the sides of a triangle.

Applications of the converse of Pythagoras theorem:

Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.

 

Questions to be Solved:

Question 1) Check whether the triangle with the side lengths, 5, 7 and 9 units is an acute ,right or an obtuse triangle.

Answer) The longest side of the triangle has a length equal to 9 units. Now, compare the square of the length of the longest side with the sum of squares of the other two sides that have been given.

Square of the length of the longest side is 9² = 81 units. 

Sum of the squares of the other two sides is equal to,

5² + 7² is equal to 25+ 49 which is equal to 74 square units.

This concludes that, 9² > 5² + 7²

Therefore, by the corollary to the converse of the Pythagorean Theorem, the triangle is an obtuse triangle.

Question 2) Check whether a triangle with side lengths 6 cm, 10 cm, and 8 cm is a right triangle. Check whether the square of the length of the longest side is the sum of the squares of the other two sides.

Answer) Apply the converse of Pythagorean Theorem.

(10)² = (6)² + (8)²

100 = 36 + 64

Since the square of the length of the longest side is the sum of the squares of the other two sides, by the converse of the Pythagorean Theorem, the triangle is referred to as a right triangle.

A corollary to the theorem categorizes the triangles into acute triangle, right triangle, or obtuse triangle.

In a triangle with side lengths a, b, and c where c is the length of the longest side,

If (c)² < (a)² + (b)²  then we can say that the triangle is acute, and

If (c)² < (a)² + (b)² then we can say that the triangle is obtuse.

FAQ (Frequently Asked Questions)

Question 1. What is the Difference Between the Pythagorean Theorem and the Converse of the Pythagorean Theorem?

Answer. The Converse of the Pythagorean Theorem

In mathematics a converse can be defined as a theorem whose hypothesis and conclusion are the conclusion and hypothesis of some other theorem. In other words, the converse of the Pythagorean Theorem can be known as the reversal of the Pythagorean Theorem

Question 2. What is the Converse of a Theorem?

Answer. A converse of a theorem is a statement formed by interchanging what is given in a theorem and what is to be proved. For example, the isosceles triangle theorem in Mathematics states that if two sides of a triangle are equal then the two angles are equal.

Question 3. How can you use the Converse of the Pythagorean Theorem to Tell if a Triangle is a Right Triangle?

Answer. The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is known as a right triangle.