Converse of Pythagoras Theorem

As we know, the Pythagoras (Pythagorean) Theorem states that for a right angle triangle, the square of the hypotenuse is equal to the sum of the squares of the other 2 sides.

The above relation is mathematically represented as,

a² + b² = c², c = length of the hypotenuse, a & b are the lengths of the other 2 sides

This article will deal with the converse of the Pythagorean Theorem.

Theorem

The converse of the Pythagorean Theorem states 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 2 sides, then the triangle is a right triangle.

The corollary states if a triangle has a side length of a, b, and c, where c is the hypotenuse, we can determine

It thus gives us an easy way to prove whether a triangle is a right triangle or not.

A triangle has sides 5 m and 12 m, with the longest side of 13 m. Is it a right-angle triangle?

Solution:

As we know,
In a right angle triangle,
a² + b² = c², here c = 13, a = 5 m, b = 12 m
If the given triangle is a right angle triangle, then
5² + 12² = 13² will be true
Now,
5² + 12²
=> 25 + 144
=> 169
=> 13²
Thus, 5² + 12² = 13² holds true for the Pythagorean Theorem, thus it is a right angle triangle.

Not only the Converse of the Pythagorean Theorem tells us whether a triangle is a right angle or not, its corollary states if a triangle has a side length of a, b, and c, where c is the hypotenuse, we can determine whether the triangle is acute or obtuse. This is done by comparing the sum of the squares of the shorter sides with the square of the longest side of the triangle as given below.

Thus,

• If the sum of the squares of the shorter sides is equal to the square of the longest side (a² + b² = c²), the triangle is a right triangle. This means the longest side of the triangle lies opposite to the 90°
• If the sum of the squares of the shorter sides is greater than the square of the longest side (a² + b² > c²), then the triangle is an acute triangle.
• If the sum of the squares of the shorter sides is smaller than the square of the longest side (a² + b² < c²), then the triangle is an obtuse triangle.

Determine whether a triangle with sides 3 cm, 5 cm, and 7 cm is an acute, right or obtuse triangle.

Solution:

As we know,
In a right angle triangle,
a² + b² = c², here a = 3 cm, b = 5 cm, c = 7 cm
Now,
a² + b² = 3² + 5² = 9 + 25 = 34
c² = 7² = 49
Thus,
36 < 49 (a² + b² < c²) and thus it is an obtuse triangle

Proof

Converse of Pythagoras Theorem

To prove:

ΔABC is a Right Triangle

Proof:

AC² = AB² + BC²
We make a right angle triangle PQR such as PQ = AB and QR = BC
Now, in right angle triangle PQR, we have:
PR² = PQ² + QR² (By Pythagoras Theorem, as ∠Q=90°)
=> PR² = AB² + BC² (By construction)…….. (1)
Again, AC² = AB² + BC² (Given)………. (2)
Thus, AC = PR [From (1) and (2)]………….. (3)
In ΔABC and ΔPQR,
AB = PQ [By construction]
BC = QR [By construction
AC = PR [Proved by in (3)]
Thus,
ΔABC ≃ ΔPQR [By SSS congruence theorem]
∠B = ∠Q [Corresponding angles of congruent triangles]
∠Q = 90° [By construction]
Hence,
∠B = 90°
Thus, ΔABC is a right triangle
Hence Proved