Drop a Perpendicular
The distance from a point to a line is measured along the perpendicular. Slide to feel it shrink to zero when the point lies on the line.
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
- Slide the line until it passes through (3, 0): when the point lies ON the line, the perpendicular distance is exactly zero.
- Keep it there and pass through (1, 2) as well — distance to THAT point is now zero too.
- Slide the line onto (4, 0). A point ON the line is exactly zero distance from it.
- Keep it there and take (0, 6) too — now both distances are zero at once.
Check yourself
The distance from a point P to the line ax + by + c = 0 is |a*Px + b*Py + c| / sqrt(a^2 + b^2). Why divide by sqrt(a^2 + b^2)?
- To normalise the line coefficient vector; without it, scaling the equation (e.g. multiplying by 2) would change the apparent distance. — correct
- Because Pythagoras theorem always involves a square root.
- To make the answer positive.
- Because the formula is just defined that way and you have to memorise it.
Think about it
- Where on the line is the closest point to (3, 0)? Not directly above — drop a perpendicular.