Long before Pythagoras, the ancient Indian mathematician Baudhayana stated in the Sulba Sutras (around 800 BCE): "The diagonal of a rectangle produces both areas which its length and breadth produce separately." This is precisely the Pythagorean theorem. This chapter explores this profound relationship and its applications.
The Theorem
In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.
If the legs are a and b, and the hypotenuse is c, then: c2 = a2 + b2
Converse: If in a triangle c2 = a2 + b2, then the angle opposite c is a right angle.
Pythagorean Triplets: Sets of three whole numbers (a, b, c) satisfying a2 + b2 = c2.
Common triplets: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25).
Any multiple of a triplet is also a triplet: e.g., (6, 8, 10).
A right triangle has legs 6 cm and 8 cm. Find the hypotenuse.
c2 = 62 + 82 = 36 + 64 = 100.
c = 10 cm. (6, 8, 10) is a Pythagorean triplet.
Is (5, 12, 13) a Pythagorean triplet?
52 + 122 = 25 + 144 = 169 = 132. Yes, it is.
A ladder 13 m long leans against a wall. Its foot is 5 m from the wall. How high does it reach?
Height h: h2 + 52 = 132 → h2 = 169 - 25 = 144 → h = 12 m.
Check whether a triangle with sides 7 cm, 24 cm, 25 cm is right-angled.
72 + 242 = 49 + 576 = 625 = 252. Yes, it is right-angled.
Find the diagonal of a rectangle with length 15 cm and width 8 cm.
diagonal2 = 152 + 82 = 225 + 64 = 289. Diagonal = 17 cm.
An equilateral triangle has side 6 cm. Find its height.
Drop a perpendicular from one vertex. It bisects the opposite side.
h2 + 32 = 62 → h2 = 36 - 9 = 27 → h = 3√3 cm ≈ 5.2 cm.
A square has a diagonal of 10 cm. Find its side.
Let side = a. a2 + a2 = 102 → 2a2 = 100 → a2 = 50 → a = 5√2 cm ≈ 7.07 cm.
- Key formulas:
- Hypotenuse: c = √(a2 + b2)
- Leg: a = √(c2 - b2)
- For a square of side s: diagonal = s√2
- For an equilateral triangle of side s: height = (s√3)/2
Common mistakes
- Confusing the hypotenuse with a leg — the hypotenuse is always the longest side, opposite the right angle.
- Forgetting that the converse also holds: you can test whether a triangle is right-angled using the theorem.
- Not simplifying surds properly (e.g., √27 = 3√3, not just √27).
Summary
The Baudhayana–Pythagoras theorem is one of the most useful tools in geometry. It applies to any right-angled triangle and allows us to find unknown sides, check for right angles, and solve real-world problems involving distances and heights.