m = (y2 - y1) / (x2 - x1)
now, if 2 lines are perpendicular, then the product of their slopes is -1 what is the slope of x -y = 5 ?
to find out, rewrite it in slope - intercept form : y = mx + n, with m = slope and n = intercept
so x - y = 5 <=> y = x - 5
hence, its slope is m = 1
then, what's the slope of ANY line perpedicular to it ? let M be the slope of that line
as we said above: m * M = -1
with m = 1 => M = -1
coming back to the definition of the slope, given 2 points on it
(y2 - y1) / (x2 - x1) = M
we know y2 = 3 and x2 = 2 AND M = -1
replace them, and what do u get u think ? the eqn of the line passing through (2,3) and is perpendicular to x - y = 5
hence, (3 - y)/(2 - x) = -1 => y - 3 = 2 - x => y = -x + 5
once we know its equation, how do we find the distance between that point on it (2,3) and x - y = 5 ?
1stly, we find the intersection point of the new line and the one given; to do this, we need to solve the linear system of the their equations
x - y = 5
x + y = 5
add 1st to 2nd
2x = 10 => x = 5 => y = 0
hence, the intersection between the given line and the perpendicular to it passing through (2,3) is (5,0)
