Over 2,500 years ago, the Greek mathematician Pythagoras proved one of the most useful relationships in all of mathematics. Today, that same relationship helps engineers build bridges, architects design buildings, and students crack board exam questions. Let us explore the Pythagorean theorem from first principles.
What is the Pythagorean Theorem?
In any right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides:
a² + b² = c²
Where c is the hypotenuse and a, b are the two shorter sides (called legs or perpendicular sides).
Proof of the Pythagorean Theorem
There are over 370 known proofs. The simplest one for Class 9:
- Draw a square with side (a + b).
- Inside it, arrange four identical right-angled triangles with legs a and b, leaving a square of side c in the middle.
- Area of big square = (a + b)² = a² + 2ab + b²
- Area of 4 triangles = 4 × ½ab = 2ab
- Area of inner square = c²
- So: a² + 2ab + b² = 2ab + c² → a² + b² = c² ✓
Pythagorean Triplets
Some sets of three whole numbers perfectly satisfy a² + b² = c². These are called Pythagorean triplets. The most common ones to memorise:
| a | b | c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
Multiples also work: 6, 8, 10 (double the 3-4-5 triplet) and 9, 12, 15, etc.
Step-by-Step Solved Examples
Example 1: A ladder 10 m long leans against a wall. Its foot is 6 m from the base of the wall. How high up the wall does the ladder reach?
Here c = 10 (ladder), a = 6 (distance from wall), b = ?
b² = 10² − 6² = 100 − 36 = 64
b = 8 m ✓
Example 2: A cricket pitch is rectangular, 20 m long and 3 m wide. What is the length of the diagonal?
Diagonal² = 20² + 3² = 400 + 9 = 409
Diagonal = √409 ≈ 20.22 m ✓
Common Mistakes
- Squaring the wrong side: c must always be the hypotenuse — the longest side. Never write a² = b² + c².
- Forgetting to square-root: After computing c², students sometimes report c² as the answer instead of taking √.
- Using it for non-right triangles: The theorem only applies to right-angled triangles. For other triangles, use the Cosine Rule.
Real-World Applications in India
- Construction: Builders use the 3-4-5 triplet to check that a corner is exactly 90° without a protractor.
- Mobile towers: Engineers calculate how long a support cable needs to be based on the tower height and horizontal distance.
- GPS: Triangulation used in GPS navigation is based on extensions of the Pythagorean theorem into 3D (distance formula).
Practice Problems
- (Easy) The sides of a right triangle are 9 cm and 40 cm. Find the hypotenuse.
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c = √(81 + 1600) = √1681 = 41 cm - (Medium) Is a triangle with sides 11, 60, 61 right-angled?
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11² + 60² = 121 + 3600 = 3721 = 61². Yes! - (Hard) A pole 15 m high is broken by the wind. Its top touches the ground 9 m from its base. Find the height at which it broke.
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Let break height = h. Then (15−h)² = h² + 81. Solving: h = 9.4 m approx.
Visualise the Pythagorean theorem interactively on MathVis — drag the sliders to change the triangle sides and watch the theorem hold in real time.