📐 Pythagoras: The Secret of the Squares

This is a right-angled triangle: one corner is a right angle. The two sides that make the right angle are the legs: a and b. The side opposite the right angle, the longest side, is the hypotenuse: c.

Let’s draw a square on each side of the triangle. The squares on the legs have 3 × 3 = 9 and 4 × 4 = 16 unit squares. The square on the hypotenuse has 5 × 5 = 25 unit squares.

Let’s take the 9 and 16 unit squares from the small squares and fit them into the big square. They all fit exactly, with no gaps! 9 + 16 = 25. The areas of the squares on the legs add up to the area of the square on the hypotenuse.

Here is Pythagoras’ theorem: in a right-angled triangle, a² + b² = c². The squares of the two legs add up to the square of the hypotenuse. In our triangle, 3² + 4² = 9 + 16 = 25 = 5².

A ladder leans against a wall. Its foot is 6 m from the wall, and it reaches 8 m up the wall. The ladder is the hypotenuse: 6² + 8² = 36 + 64 = 100. Since c² = 100, c = 10 m.

This time we know the ladder’s length: 15 m. Its foot is 9 m from the wall. How high up the wall does it reach? 9² + b² = 15², so 81 + b² = 225. That gives b² = 144, so b = 12 m. If we know the hypotenuse, we can find a leg too.

We can also find out whether a triangle has a right angle. We look at the square of the longest side. 8, 15, 17: 8² + 15² = 64 + 225 = 289 = 17². They match, so it is a right-angled triangle! 4, 5, 6: 4² + 5² = 41 but 6² = 36. They don’t match, so it is not a right-angled triangle.

Long ago, some builders used a rope with 12 knots to make a right angle. The knots are evenly spaced. Count 3 spaces along one side, 4 along another and 5 along the third, and pull the rope tight to make a triangle. Because 3² + 4² = 5², there is a right angle between the short sides.

Summary: in a right-angled triangle, the squares of the legs add up to the square of the hypotenuse: a² + b² = c². We can use it to find a ladder’s length, a missing side, and whether a triangle has a right angle. Now try it in Pythagoras Ladder!