Converse of Pythagoras Theorem

Converse of the Pythagorean Theorem helps identify if a triangle is a right triangle. This theorem also helps determine if the triangle is acute or obtuse, making it a renowned principle in trigonometry.

Definition

• A triangle is a right triangle if the sum of the squares of its two shorter sides equals the square of the longest side (hypotenuse).
• An obtuse triangle has a sum of squares of its two shorter sides less than the square of the longest side (hypotenuse).
• An acute triangle has a sum of squares of its two shorter sides greater than the square of the longest side (hypotenuse).

Notes:

1. The sum of angles in any triangle always equals 180°.
2. In a right triangle, one angle measures 90°, with the other two angles summing up to 90°.

Proof of Converse of Pythagoras Theorem

The Pythagoras theorem states that if a triangle is right-angled (90°), then the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. 

So, In the given triangle ABC (Fig.-1), we have BC² = AB² + AC². Here, AC is the base, AB is the altitude or the height, and BC is the hypotenuse. 

If this is true then we need to prove that ∠A = 90°

To start with, we construct a XYZ right-angled at X such that XY = AB and XZ = AC.

In XYZ, by Pythagoras Theorem:

YZ² =XZ² + XY² = b² + c² …..(1)

In ABC, by Pythagoras Theorem:

BC² =AC² + AB² = b² + c² …..(2)

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

YZ² = BC²

YZ = BC

 ACB = XYZ (By SSS congruence)

∠A = ∠X (Corresponding angles of congruent triangles)

∠x = 90° (By construction)

So ∠A = 90°.

Hence, the converse of the Pythagoras theorem is proved.

Important Formulas

• If the three sides of a triangle are a,b & c. and, b2 +c2=a2 (a is the longest side of the triangle known as hypotenuse) 
• If the three sides of a triangle are a,b & c. And, b2 +c2 < a2 (a is the longest side of the triangle known as hypotenuse) 
• If the three sides of a triangle are a,b & c. And, b² +c² > a² (a is the longest side of the triangle known as hypotenuse). 

Solved Examples

Example.1 The sides of a triangle are 3, 4 and 5. Check whether the given triangle is a right triangle or not.

Solution;

Let, 

a = 5 (Longest Side)

b = 4

c = 3

Apply the converse of Pythagoras' Theorem

For a Right Triangle

b² +c² = a²

or

a² = b² +c²

Put the values in the above equation

42 + 32 = 52

16 + 9 = 25

25 = 25

So, the given side satisfied the right triangle condition.

Hence, this triangle will be a right-angle triangle.

Example.2 If the triangle has side lengths of 5, 7, and 9 units then it will be a right angle triangle or not?

Solution;

The longest side of the triangle is 9 units.

Now compare the square of the length of the longest side of the triangle and the sum of squares of the other two sides.

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

Sum of the squares of the other two sides is

52 +72=25+49 =74 sq. units.

So. 52 +72 < 92

Therefore, by the converse of Pythagorean Theorem, the triangle is not a right triangle.