Section Formula

1. Introduction: Finding the point that splits a line in a given ratio

In the previous topic you learned to find the distance between two points. You could take two dots on a grid and say exactly how far apart they are.

Now we ask a different question. Suppose you have two points joined by a straight line. There is a third point sitting on that line, somewhere between the two ends. Where exactly is it?

Here is the everyday picture. Think of a rope tied between two poles in a field. A bird lands on the rope, not in the middle, but closer to one pole than the other. You want to tell someone the bird's exact position. The section formula is the tool that does this. It finds the coordinates of a point that divides a line segment in a given ratio.

A small piece of vocabulary first. A ratio like $2:3$ means "2 parts to 3 parts." It is just a way of saying how something is split. If a rope is cut in the ratio $2:3$, the first piece has 2 parts and the second piece has 3 parts, so 5 equal parts in total.

Stop scrolling. If a $10$ m rope is cut in the ratio $2:3$, how long is the first piece? Try it in your head before reading on.

(One part is $10 \div 5 = 2$ m. The first piece is $2 \times 2 = 4$ m. The second is $3 \times 2 = 6$ m.)

You can now picture what "divides in a ratio" means. Next we find the actual coordinates.