Internal Division: Cutting a Line in a Ratio
A point dividing PQ internally in ratio m:n has coords ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)).
Equation
y = x
Graph
Table
| x | y |
|---|---|
| -5 | -5 |
| -4 | -4 |
| -3 | -3 |
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
What this lesson covers
What you do
You shape the function y = m*x + c and watch the graph answer.
Challenges to clear
- Pass through the point dividing (0,0) and (10,5) internally in ratio 2:3 — the point (4, 2).
- Now the 1:4 point of the SAME segment: (2, 1). One line y = x/2 carries every division point.
- The point dividing (0, 0) and (6, 9) in ratio 2:1 is (4, 6). Send the line through it.
- Now the 1:2 point of the SAME segment: (2, 3). Every division point of a segment lies on one line.
Check yourself
For internal division of PQ in ratio m:n, the section point formula is:
- ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)) — weighted average where the FAR endpoint coordinate gets the m weight. — correct
- ((nx1 + mx2)/n, (ny1 + my2)/n) — no denominator m+n.
- ((mx1 + nx2)/(m+n), ...) — but with x1 getting the m weight reverses the ratio.
- ((x1+x2)/m, (y1+y2)/n) — wrong shape entirely.
Think about it
- Where does the point dividing (0,0) to (10,5) in ratio 2:3 land? Predict, then make the line pass through it.