Optimization Problem - Shortest distance from a line to a point

Question

“Use mathematical optimization to prove that the shortest distance from a fixed point A to any point on a given line is the length of the line segment joining the point A to the line and is perpendicular to the line.”

I’m assuming you’d have to find the derivative of the distance formula but I’m a bit lost. I tried letting A equal (x°, y°) and in the end I got y=-1/m (x-x°) + 3y°/m but I feel like this is not correct. How does one go about proving this?

Answer 1

I don't know what "mathematical optimization" means in this context.

The easiest way to prove this statement is to draw the perpendicular from point A to the line.

Then select any other point on the line and draw the line from point A to this selected point..

You have drawn a right triangle in which the perpendicular is a leg and the segment to any other point is the hypotenuse. The Pythagorean theorem tells you that the hypotenuse is always longer!