It looks hard, but take it step by step:
1st thing to know is to make right angles (90°), the slopes of two lines making angle must be perpendicular. And perpendicular lines have slopes that are negative reciprocals.
For AB, slope is (y2-y1)/(x2-x1) = (-1-2)/(-3-1)
= -3/-4 = 3/4 --> neg recip is -4/3
When C (-3, -2), AC slope is (-2-2)/(-3-1)
= -4/-4 = 1
BC = (-2--1)/(-3--3) = -1/0 = undefined
No right angles, bc no lines = -4/3
When C (0, -5), AC slope is (-5-2)/(0-1)
= -7/-1 = 7
BC = (-5--1)/(0--3) = -4/3
So BC is perpendicular to AB when C (0,-5)
For C (1, -2), AC slope is (-2-2)/(1-1)
= -4/0 = undefined
BC slope = (-2--1)/(1--3) = -1/4
both not = -4/3, so no right angles
Therefore only when C = (0, -5) will ABC be a right triangle
