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Pythagorean Theorem

🎯 Learning Goals

  • Understand the relationship between the three sides of a right-angled triangle
  • Use the theorem to calculate unknown distances in the real world

💡 Why Learn This?

The Pythagorean Theorem is one of the most famous and useful formulas in mathematics. It is used in construction, navigation, computer graphics, and even calculating the shortest path in GPS systems!

a² + b² = c²

In any right-angled triangle, the square of the longest side (the hypotenuse) is exactly equal to the sum of the squares of the other two sides. This means if you know the lengths of any two sides, you can always find the third!

abc

How to Use It

  • ・ Finding the hypotenuse (c): If a = 3 and b = 4, then 3² + 4² = 9 + 16 = 25. Since c² = 25, c = 5.
  • ・ Finding a shorter side (a): If c = 13 and b = 12, then a² + 12² = 13², so a² + 144 = 169. a² = 25, therefore a = 5.
  • ・ Pythagorean Triples: Sets of integers like (3,4,5) and (5,12,13) that perfectly fit the theorem without decimals.

⚠️ Common Pitfalls

A common mistake is forgetting to take the square root at the very end. Finding c² = 25 does NOT mean the side length is 25. You must find the square root, so c = 5!

Interactive Pythagorean Visualizer

Adjust the sides 'a' and 'b' to see how the hypotenuse 'c' changes dynamically!

a = 3b = 4c = 5

Drag the sliders to change the base (a) and height (b):

3
4
Equation:
3² + 4² = c²
9 + 16 = 25
c = √25 ≈ 5

📝 Summary & Recap

  • The theorem only works for right-angled triangles (triangles with a 90° angle).
  • The formula is a² + b² = c², where 'c' is the longest side across from the right angle.

Quick Drill

Test your understanding of the Pythagorean Theorem!

In a right-angled triangle, what is the name of the longest side (c)?

🔍 Deep Dive (Optional)

While named after the Greek mathematician Pythagoras, evidence shows that ancient Babylonians, Indians, and Chinese mathematicians understood and used this relationship more than a thousand years before Pythagoras was even born!

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