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Pythagoras Theorem

Distance Between Two Points

We use the Pythagoras Theorem to derive a formula for finding the distance between two points in 2- and 3- dimensional space.

Let P = (x ₁, y ₁) and Q = (x ₂, y ₂) be two points on the Cartesian plane (see picture below).

So the horizontal distance between P and Q is x2 - x1 and the vertical distance between the points is y2 - y1.

Then from the Pythagoras Theorem we find that the distance between P and Q is

In a similar way, it can be proved that if P = (x ₁, y ₁, z₁) and Q = (x ₂, y ₂, z₂) are two points in the 3-dimensional space, the distance between P and Q is

the square root of the sum of the squares of each of the x distance between the points, the y distance between the points and the z distance between the points.

Exercise 1.

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