Foot of Perpendicular Formula (2D Coordinate Geometry)

• Let the given straight line be the equation: $Ax + By + C = 0$.
• Let the given floating point in space be: $P(x_1, y_1)$.
• Let the unknown 'Foot' (the point on the line where they intersect at 90 degrees) be: $Q(h, k)$.

To directly find the coordinates $(h, k)$ without solving long simultaneous equations, you can use this brilliant shortcut formula:

$\frac{h - x_1}{A} = \frac{k - y_1}{B} = - \frac{(Ax_1 + By_1 + C)}{A^2 + B^2}$

1. First, plug the coordinates of the floating point $(x_1, y_1)$ directly into the line equation ($Ax + By + C$) to find the value of the numerator on the right side.
2. Calculate the denominator ($A^2 + B^2$).
3. This entire right-hand side will now become a single, simple number (let's call it $M$).
4. Now, separate the equation into two easy parts:
   • $\frac{h - x_1}{A} = M$ (Solve this to find the $h$ coordinate).
   • $\frac{k - y_1}{B} = M$ (Solve this to find the $k$ coordinate).
5. Your final answer is the point $(h, k)$.

A very similar question asks you to find the 'Image' (or Reflection) of the point, pretending the line is a mirror. The formula is almost exactly the same, but you multiply the right side by 2:

$\frac{h - x_1}{A} = \frac{k - y_1}{B} = - 2 \frac{(Ax_1 + By_1 + C)}{A^2 + B^2}$