AI Tutor
If you walk 4 blocks east and 3 blocks north on a city grid, you walk 7 blocks in total. But what if a drone flies straight from your start to your finish?
The straight-line path cuts diagonally across the grid. To find that exact diagonal length, we turn the grid into a right triangle.

Building the Right Triangle

On a Cartesian grid, any two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) form the opposite corners of a right triangle.
The hypotenuse is the longest side of a right triangle, located directly across from the 90∘90^\circ right angle.
Around 500 BCE, ancient Greek mathematicians formalized the Pythagorean theorem, which proves that the sum of the squared legs equals the squared hypotenuse: a2+b2=c2a^2 + b^2 = c^2.
How do we turn this geometry rule into an algebraic formula for any pair of coordinates?

Deriving the Distance Formula

The horizontal leg length is the difference between xx-values, Ξ”x=x2βˆ’x1\Delta x = x_2 - x_1. The vertical leg length is the difference between yy-values, Ξ”y=y2βˆ’y1\Delta y = y_2 - y_1.
Plugging these legs into the Pythagorean relationship gives d2=(x2βˆ’x1)2+(y2βˆ’y1)2d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2. Taking the principal square root gives the distance formula:
d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
Let's test this formula on a concrete pair of coordinates.