What is Pythagoras' Theorem?
Over 2,500 years ago, a Greek mathematician named Pythagoras discovered a remarkable relationship between the sides of a right-angled triangle. This relationship is one of the most famous and useful results in all of mathematics.
In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Written algebraically: where is the hypotenuse and and are the other two sides.
The longest side of a right-angled triangle, always located directly opposite the right angle (90°).
The theorem only works for right-angled triangles — triangles that contain exactly one 90° angle. Before applying the theorem, always check that you're dealing with a right angle!
Imagine you're standing at one corner of a rectangular football pitch. You could walk along two edges to reach the opposite corner, or you could cut diagonally straight across. Pythagoras' Theorem tells you exactly how long that diagonal shortcut is, based on the length and width of the pitch.
Understanding the Theorem Visually
The equation has a beautiful geometric meaning. If you draw a square on each side of a right-angled triangle:
- The square on side has an area of
- The square on side has an area of
- The square on the hypotenuse has an area of
Pythagoras' Theorem says that the area of the two smaller squares added together equals the area of the largest square.
For example, in a triangle with sides 3, 4, and 5:
- Area of square on side 3 =
- Area of square on side 4 =
- Area of square on side 5 =
- And indeed: ✓
This visual proof helps you understand why the theorem works, not just how to use it. In MYP Sciences, understanding the reasoning behind formulas is just as important as applying them.
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