Geometry of the Line (3): Length

It is a fairly simple job to calculate the length of a line. So far I've shown the mid-points, gradients, and equations. This is similar stuff, and based on the principle of Pythagoras, as for gradients.

Take a line with two points (3,6) and (8,12) lying on the line.

We can see any straight line on a graph as being equivalent to a hypotenuse of a right angled triangle.

Now Pythagoras says that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Looking at the diagram:




The line we want to measure is the hypotenuse and the other two sides are clearly horizontal and vertical. So the length of the horizontal line is easy - just x₂ - x₁ and the length of the vertical line the same but with ys. y₂-y₁

This is where Pythagoras comes in. We square the lengths of the two lines and add them together. This gives us the square of the hypotenuse.

We write it like this:

√(x₂-x₁)² + (y₂-y₁)²

In this case:

√ (8-3)² + (12-6)²

√ 25 +36

√ 61

This will be a surd so we can leave the answer as √61 for the purposes of C1.