Pythagoras Theorem and Its Converse

There is a fixed relationship among the lengths of the three sides of every right-angled triangle. This relationship is called the Pythagoras Theorem. It helps us find an unknown side of a right triangle. Its converse also helps us decide whether a triangle is right-angled when its three side lengths are known.

📚 Understanding the relation through areas

Suppose the two perpendicular sides of a right triangle have lengths a and b, and the hypotenuse has length c.

If squares are drawn on these three sides, their areas are respectively

a², b² and c².

The Pythagorean relation is

c² = a² + b².

In words,

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

📚 A simple example

If the perpendicular sides of a right triangle are 3 units and 4 units, and the hypotenuse is 5 units, then

25 = 9 + 16.

So the areas of the three squares satisfy the same relation.

📚 An area-based verification

Take two congruent right triangles whose perpendicular sides are a and b, and whose hypotenuse is c. Arrange them to form a trapezium.

The parallel sides of the trapezium have lengths a and b, and the distance between them is a + b. Therefore,

Area = 1/2 × (a + b) × (a + b).

The same region can also be divided into two right triangles and the middle triangle. Therefore,

1/2(a + b)² = 1/2 ab + 1/2 ab + 1/2 c².

Multiplying both sides by 2,

(a + b)² = 2ab + c².

So,

a² + 2ab + b² = 2ab + c².

Hence,

a² + b² = c².

This area interpretation makes the basic Pythagorean relation easy to visualise.

⭐ Theorem 49: Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

📚 General form of the theorem

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

c² = a² + b².

Therefore,

c = √(a² + b²).

To find an unknown perpendicular side,

or

b = √(c² - a²).

📚 Historical note: Baudhāyana

Ancient Indian mathematical texts associated with Baudhāyana contain a statement equivalent to the Pythagorean relation for the diagonal of a rectangle. This work predates Pythagoras by several centuries and is an important part of the history of geometry in India.

⭐ Theorem 50: Converse of Pythagoras Theorem

If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

📚 Identifying a right triangle using the converse

Let the largest side of a triangle be c and the other two sides be a and b. Check whether

c² = a² + b².

If the equality is true, the angle opposite the largest side c is 90°, so the triangle is right-angled.

Example: 5, 12, 13

and

5² + 12² = 25 + 144 = 169.

Therefore, 13² = 5² + 12². Hence the triangle is right-angled, and the side of length 13 is the hypotenuse.

📚 Generating Pythagorean triples

The algebraic identity

shows that, for m > n,

m² - n², 2mn and m² + n²

can be used as the side lengths of a right triangle. The hypotenuse is

m² + n².

Example 1

For m = 2, n = 1,

m² - n² = 3, 2mn = 4, m² + n² = 5.

So 3, 4, 5 is a Pythagorean triple.

Example 2

For m = 3, n = 2,

m² - n² = 5, 2mn = 12, m² + n² = 13.

So 5, 12, 13 is another Pythagorean triple.

📚 Solved examples

Example 3: Find the hypotenuse

The perpendicular sides of a right triangle are 6 cm and 8 cm. Find its hypotenuse.

Therefore,

c = 10 cm.

Example 4: Find an unknown side

The hypotenuse of a right triangle is 17 cm and one perpendicular side is 8 cm. Find the other side.

Let the unknown side be x.

Therefore,

x = 15 cm.

Example 5: Is the triangle right-angled?

Check a triangle with sides 7 cm, 24 cm and 25 cm.

The largest side is 25.

and

7² + 24² = 49 + 576 = 625.

Therefore, the triangle is right-angled.

✍️ Practice problems

1. A right triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse.

2. The hypotenuse of a right triangle is 13 cm and one perpendicular side is 5 cm. Find the other side.

3. Check whether a triangle with sides 8 cm, 15 cm and 17 cm is right-angled.

4. Check whether a triangle with sides 6 cm, 8 cm and 11 cm is right-angled.

5. A rectangular field is 24 m long and 10 m wide. Find the length of its diagonal.

6. Use m = 4 and n = 1 to generate a Pythagorean triple.

⚠️ Common mistakes

👉 Mistake: Using a² + b² = c² for every triangle.
Correct: Pythagoras theorem applies directly only to a right-angled triangle.

👉 Mistake: Taking any side as c.
Correct: c must be the side opposite the right angle, that is, the hypotenuse.

👉 Mistake: Adding squares while finding an unknown perpendicular side when the hypotenuse is known.
Correct: Square of the unknown perpendicular side = square of the hypotenuse − square of the other perpendicular side.

👉 Mistake: Testing the square of the smallest side when three side lengths are given.
Correct: In the converse theorem, compare the square of the largest side with the sum of the squares of the other two sides.

👉 Mistake: Treating 5² + 12² = 13² as only a numerical coincidence.
Correct: The equality allows us to conclude, by the converse theorem, that the triangle is right-angled.

📚 Important points to remember

👉 In a right triangle, the side opposite the right angle is the hypotenuse, and it is the longest side.

👉 If ∠A = 90°, then BC² = AB² + AC².

👉 In general, c² = a² + b².

👉 Unknown hypotenuse: c = √(a² + b²).

👉 Unknown perpendicular side: a = √(c² - b²) or b = √(c² - a²).

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

👉 3, 4, 5, 5, 12, 13 and 8, 15, 17 are common Pythagorean triples.

👉 (m² - n²), 2mn, (m² + n²) generates a Pythagorean triple when m > n.