How do you use Pythagoras' Theorem in three dimensions?

You use Pythagoras' Theorem in three dimensions to find the distance between two points in 3D space.

In three dimensions, Pythagoras' Theorem helps you calculate the length of the diagonal line connecting two points, \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\). To do this, you extend the two-dimensional Pythagorean Theorem into three dimensions. The formula you use is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

Here, \(d\) represents the distance between the two points. The terms \((x_2 - x_1)\), \((y_2 - y_1)\), and \((z_2 - z_1)\) are the differences in the x, y, and z coordinates of the two points, respectively.

Imagine you have a cuboid (a 3D rectangle) and you want to find the length of the diagonal that stretches from one corner of the cuboid to the opposite corner. First, you find the diagonal in the base of the cuboid using the 2D Pythagorean Theorem:

\[ \text{Base diagonal} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Next, you treat this base diagonal as one side of a right-angled triangle, with the height of the cuboid being the other side. You then apply Pythagoras' Theorem again in three dimensions:

\[ d = \sqrt{(\text{Base diagonal})^2 + (z_2 - z_1)^2} \]

By substituting the base diagonal back into the formula, you get the full 3D distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]

This method allows you to find the straight-line distance between any two points in three-dimensional space, making it a powerful tool in geometry and physics.