The eq of the line perpendicular to the line y=4/3x-100 (1) and passing through (0,0) will be y=-3/4x (2)

If the eq of a line is of the form y=mx+c , where m is the slope and c is the y intercept. The line perpendicular to it will have the slope -1/m. Simply the product of two slope = -1

The eq of line perpendicular to y = 4/3x - 100 should be y=- 3/4x + c . but it has a value (0,0) so c = 0.

solving eg (1) and (2) we get 4/3x + 3/4 x = 100 , x = 48 and -36
The point where the two lines meet and are pendicular to each other.

The distance between (0,0) and (48,-36) is shortest distance
square root of (48-0)²+(-36-0)² = 60