Pythagoras

1) Hook — A Fun Real-Life Story

Imagine you are at a cricket ground in Mumbai, watching your favourite batsman hit a perfect cover drive. The ball travels in a straight line from the bat to the boundary, but the fielder runs diagonally across the field to catch it. How can the fielder figure out the shortest path to reach the ball quickly? This is where the magic of Pythagoras’ Theorem comes in!

Just like our fielder uses the shortest distance (the diagonal), Pythagoras helps us find the length of the diagonal side in a right-angled triangle, which appears everywhere — from cricket fields to building stairs in your home.

2) Core Concepts — Understanding Pythagoras’ Theorem

Pythagoras’ Theorem is a fundamental rule in geometry that applies to right-angled triangles only.

It states:

Let's name the sides:

• Hypotenuse (c): The longest side opposite the 90° angle.
• Base (a): One side adjacent to the right angle.
• Perpendicular (b): The other side adjacent to the right angle.

Mathematically,

Example 1: Finding the Hypotenuse

A ladder leans against a wall. The base of the ladder is 3 metres away from the wall, and the ladder reaches 4 metres high on the wall. How long is the ladder?

Here, base = 3 m, perpendicular = 4 m, hypotenuse = ?

Using Pythagoras’ Theorem:

c² = a² + b² = 3² + 4² = 9 + 16 = 25

So, c = √25 = 5 metres.

Answer: The ladder is 5 metres long.

Example 2: Finding a Missing Side

In a cricket pitch, the distance between the stumps (base) is 22 yards. If the diagonal distance from one stump to the opposite crease is 26 yards, find the length of the perpendicular side.

Given: c = 26 yards, a = 22 yards, b = ?

Using the formula:

b² = c² - a² = 26² - 22² = 676 - 484 = 192

b = √192 ≈ 13.86 yards

Answer: The perpendicular side is approximately 13.86 yards long.

3) Key Formulas / Rules

4) Did You Know?

Though Pythagoras was a famous Greek mathematician, the concept of the theorem was known and used in ancient India over 3000 years ago! Indian mathematicians called it the “Bhāvanā” and used it to build temples and palaces with perfect right angles. So, next time you see a beautiful Indian monument like the Qutub Minar or Hawa Mahal, remember Pythagoras’ Theorem helped make those straight, strong corners possible!

5) Exam Tips

• Always identify the right angle before applying the theorem.
• Remember, hypotenuse is always the longest side and opposite the right angle.
• Use a calculator carefully to find square roots; round off answers as per the question instructions.
• Common mistake: Mixing up the sides or applying the formula to non-right-angled triangles.
• Questions often ask to find the missing side or check if a triangle is right-angled by verifying if a² + b² = c².
• Board exam pattern: Usually 2–3 marks for direct application, sometimes with word problems involving real-life contexts like ramps, ladders, or sports fields.