Pythagoras Theorem and Its Converse

The side lengths of every right triangle satisfy a fixed relationship, known as Pythagoras theorem. We can use it to find an unknown side. Its converse lets us decide whether a triangle is right-angled from its three side lengths.

Pythagoras theorem

📚 Pythagoras’ relationship in terms of areas

Let the perpendicular sides of a right triangle have lengths a and b, and its hypotenuse have length c. Squares drawn on these sides have areas a², b², c² respectively. Pythagoras’ relationship is:

Area of the square on the hypotenuse = sum of the areas of the squares on the other two sides.

📚 A simple example

For a right triangle with perpendicular sides 3 and 4 units and hypotenuse 5 units:

The areas of the squares on the three sides therefore satisfy the same relationship.

📚 Proof of Pythagoras theorem using areas

Arrange two congruent right triangles to form a quadrilateral. Each triangle has perpendicular sides a and b and hypotenuse c. All lengths are positive and measured in the same unit.

Construction

🔹 1. Take collinear points A, E, D in that order, with AE = b and ED = a. Thus AD = a + b.

🔹 2. At A and D, draw perpendiculars on the same side of AD. Mark AB = a and DC = b. Hence AB ∥ DC.

🔹 3. Join BE, CE and BC. Quadrilateral ABCD is divided into △ABE, △BEC, △ECD.

Why is the middle triangle right-angled?

In △ABE and △DEC, AB = DE = a, AE = DC = b and ∠BAE = ∠EDC = 90°. By SAS congruence, △ABE ≅ △DEC and BE = EC = c.

Congruence gives ∠DEC = ∠ABE. In right triangle △ABE:

Therefore ∠AEB + ∠DEC = 90°. Since A, E, D are collinear and EB and EC lie on the same side of AD:

Hence ∠BEC = 180° − 90° = 90°. The area of the middle triangle △BEC is therefore .

Comparing areas

In ABCD, the parallel sides are AB = a and DC = b, and their perpendicular separation is AD = a + b. Thus:

Adding the areas of the three triangles gives:

Multiplying both sides by 2:

Therefore a² + b² = c². [Proved]

When a ≠ b, the quadrilateral is a trapezium. When a = b, it is a rectangle; the same area calculation still applies.

⭐ Proof of Pythagoras theorem using similar triangles

In a right triangle, the area of the square on the hypotenuse equals the sum of the areas of the squares on the other two sides.

Proof of Pythagoras theorem

📚 General form of the theorem

If c is the hypotenuse and a and b are the other two sides, then:

To find an unknown leg:

or

📚 Historical context: Baudhayana

The Sulbasutras of ancient India contain applications of Pythagoras’ relationship. A notable rule discussed in connection with the Baudhayana Sulbasutra states that the square on the diagonal of a square has twice the area of the original square. This is a special case of Pythagoras’ relationship.

Source: University of St Andrews, MacTutor: The Indian Sulbasutras.

⭐ Converse of Pythagoras theorem

If the area of the square on one side of a triangle equals the sum of the areas of the squares on its other two sides, the angle opposite the first side is a right angle.

Converse of Pythagoras theorem

📚 Identifying a right triangle using the converse

Given three lengths, first check that they are positive and that the sum of the two shorter lengths exceeds the longest. Only then can they form a triangle. Use the same unit for all lengths.

Let c be the longest side and a and b the other sides. Check whether c² = a² + b². If so, the angle opposite c is 90° and the triangle is right-angled.

Example: 5, 12, 13

Thus 13² = 5² + 12². By the converse, the triangle is right-angled and its 13-unit side is the hypotenuse.

📚 Pythagorean triples

Three positive integers a, b and c form a Pythagorean triple if a² + b² = c².

A formula for generating triples, with proof

Let m and n be positive integers with m > n. Set:

All three numbers are positive integers. Also:

These numbers can form a triangle: c is the largest and

Thus a + b > c. By the converse of Pythagoras theorem, the triangle is right-angled and its hypotenuse is m² + n².

Remember: m > n alone is not enough. In this formula for triples, m and n must be positive integers.

Example

Taking m = 2, n = 1:

Therefore 3, 4, 5 is a Pythagorean triple.

Example

Taking m = 3, n = 2:

Therefore 5, 12, 13 is another Pythagorean triple.

⚠️ Common mistakes

❌ Using a² + b² = c² for every triangle.
✔✔ Pythagoras theorem applies directly only to right triangles.

❌ Taking any side as c.
✔✔ Here c must be the hypotenuse, opposite the right angle.

❌ Adding squares to find an unknown leg.
✔✔ Square of the unknown leg = square of the hypotenuse − square of the other leg.

❌ Comparing the square of the shortest side with the sum of the other two squares.
✔✔ For the converse test, compare the square of the longest side with the sum of the squares of the other two sides.

❌ Treating 5² + 12² = 13² as just a numerical coincidence.
✔✔ Use the converse of Pythagoras theorem to conclude that the triangle is right-angled.

🧠 Key points to remember

👉 The hypotenuse is opposite the right angle and is the longest side.

👉 If ∠A = 90°, then BC² = AB² + AC².

👉 General form: c² = a² + b².

👉 Unknown hypotenuse: c = √(a² + b²).

👉 Unknown leg: a = √(c² − b²) or b = √(c² − a²).

👉 Converse: If c² = a² + b², the angle opposite c is a right angle.

👉 3, 4, 5 and 5, 12, 13 are simple Pythagorean triples.

👉 (m² − n²), 2mn, (m² + n²) generates a Pythagorean triple when m and n are positive integers with m > n.