The Section Formula Itself
3. Time to Get Calculating!
Okay, now that we’ve got a good handle on what “m” and “n” represent, let’s see how they actually fit into the section formula. The section formula helps us find the coordinates of a point P(x, y) that divides a line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n.
The formulas are as follows: x = (mx2 + nx1) / (m + n) and y = (my2 + ny1) / (m + n). See? “m” and “n” are right there in the mix! Notice how “m” is multiplied by the coordinates of point B (x2, y2), and “n” is multiplied by the coordinates of point A (x1, y1). This is important to remember to avoid mixing things up.
For internal division, we use the formulas as they are. Both “m” and “n” are positive. But for external division, remember that either “m” or “n” will be negative. So, if point P lies outside the line segment AB and is closer to B, then “m” is positive and “n” is negative. Conversely, if P is closer to A, then “m” is negative and “n” is positive.
Let’s say you have two friends planning a trip. One lives at (1, 2) and the other at (4, 5). They want to meet at a restaurant that’s twice as close to the first friend as it is to the second. Using the section formula with m = 1 and n = 2, you can find the coordinates of the restaurant (point P) and help them decide where to eat. Practice makes perfect! So, grab some problems and start plugging in those “m” and “n” values. You’ll be a section formula pro in no time!