Question 105945
In geometry, the shortest distance between a line and a point is a perpendicular segment connecting them. By definition of a perpendicular line:
slope of the perpendicular line = negative reciprocal of slope the line perpendicular to that line

So:
{{{y = mx + b}}}
{{{y = -(1/-1)x + b}}}
{{{y = x + b)}}}
since the perpendicular line contains (0,0), the origin, you can substitute its x and y coordinates to the equation:
{{{y = -x + b}}}
{{{0 = 0 + b}}}
{{{b = 0}}}
so, the equation of the perpendicular line is:
{{{y = x}}}
since the point closest to the origin in the line is the intersection of that line to the perpendicular line, we will solve for y and x:
eq1{{{y = x}}}
eq2{{{y = -x + 4}}}
Substitue x for y in eq2:
{{{y = -x + 4}}}
{{{x = -x + 4}}}
{{{2x = 4}}}
{{{x = 2}}}
since y = x:
{{{y = 2}}}
The point closest to the origin contained in line y = -x + 4 is (2,2).