How do you use Pythagoras' Theorem to find the distance between two points?

You use Pythagoras' Theorem to find the distance between two points by treating them as coordinates on a graph.

To find the distance between two points, you first need to know their coordinates. Let's say the points are \( A(x_1, y_1) \) and \( B(x_2, y_2) \). Imagine these points plotted on a Cartesian plane. The distance between these points forms the hypotenuse of a right-angled triangle, with the horizontal and vertical distances between the points forming the other two sides.

Pythagoras' Theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, this is written as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse.

In our case, the horizontal distance between the points \( A \) and \( B \) is \( |x_2 - x_1| \) and the vertical distance is \( |y_2 - y_1| \). These distances form the two shorter sides of our right-angled triangle. According to Pythagoras' Theorem, the distance \( d \) between the points is given by:

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

This formula allows you to calculate the straight-line distance between any two points on a plane. Simply substitute the coordinates of your points into the formula, perform the arithmetic operations, and you'll have your distance. This method is particularly useful in various fields, including geometry, physics, and even navigation.