The Pythagorean Theorem is really important for finding distances in coordinate geometry.

Let’s break it down:

If you have two points in a flat space, like point A at $(x_1, y_1)$ and point B at $(x_2, y_2)$, you can think of the line between these points as the longest side, called the hypotenuse, of a right triangle.

Here’s how to do it:

1. Find the Changes:

   • First, figure out how far the points are apart horizontally. This is called the horizontal change and is found by $Δx = x_2 - x_1$.
   • Then, find out how far they are apart vertically, which is the vertical change, $Δy = y_2 - y_1$.

2. Use the Theorem:

   • To find the distance $d$ between the two points, use the Pythagorean Theorem: $d = \sqrt{(Δx)^2 + (Δy)^2}$

This formula adds up the squares of the two shorter sides of the triangle and then takes the square root. This gives you the straight-line distance between the points.

It's super helpful for different problems in geometry. It helps you see how points are related and makes math calculations much easier. Plus, it’s a cool way to link algebra and geometry together!