The Pythagorean theorem · Pulse Beat
In a right-angled triangle, the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two:
In a right-angled triangle, the square of the longest side (the hypotenuse, opposite the right angle) equals the sum of the squares of the other two:
a² + b² = c²
A 3-4-5 triangle works: 9 + 16 = 25. Carpenters use it to check a corner is square.
A picture proof: take four copies of the triangle and arrange them inside a square of side (a + b), leaving a tilted square of side c in the middle. The big square's area is (a + b)². It is also four triangles (4 × ½ab = 2ab) plus c². So (a + b)² = 2ab + c², which gives a² + 2ab + b² = 2ab + c², and therefore a² + b² = c².
To find the straight-line distance between two points on a map, use the theorem on the horizontal and vertical differences.