I used GeoGebra free online graphing calculator to illustrate this problem and I had the equations typed neatly in word mathematics, but I was unable to paste the graph nor the equations into this answer. This is my first answer since this new Expert Answer. I like that we can't paste to avoid copyright infringement, but I have yet to learn how to graph and or type math in this format. I believe I will attempt the Video answer format next. message me if you have any questions, There are other, possibly more direct, ways to reach the solution to this problem.
the y = -x + 4 tells us the line has a slope of -1, and as the plot illustrates point C (the mid-point of line segment AB) is the closest point on the line to point P.
Point V is where the vertical line from point R (2, -3) intersects the line y = -x + 4 its coordinates are
(2, f(2)) where f(2) is found by plugging 2 in for x in the line equation
y = -2 + 4, solving we find V is at (2,2)
Point H is where the horizontal line from point R intersects the line, B's y coordinate is the same as the point R, and we find its x coordinate by plugging that value (-3) in for y in the line equation and solving..
-3 = -x + 4 , solving we find H is at (7, -3)
Now, angle VRH = 90 degrees, Angles RVH & RHV both equal 45 degrees, so by the triangle inequality law the shortest distance from the line to point R is from the midpoint of line segment VH or the point marked M.
V= (2,2) & H = (7, -3) we find the midpoint M at ( 9/2, -1/2)
Now we just need to find the distance from R = (x1, y1) = (2, -3) to M = (4.5, -0.5) using the distance formula
And we find the shortest distance between the line and the point is 5 (21/2) /2
5(sqrt2)/2
