Question

The vertices of a triangle are A(–2, 2), B(9, 0), and C(1, –4). If the triangle is a right triangle, which vertex is the right angle?

Answer 1

There are three pairs of vertices:
AB
AC
BC
Use the coordinates to find the three slopes of those pairs.
Then find if any two slopes are negative reciprocals.
The slopes of perpendicular lines are negative reciprocals.

Answer 2

first, see what you get

but as mathstudent55 pointed out

two slopes will be the similar, to the extent that

one will be the other upside-down and negative

those 2 slopes, are the perpendicular ones

and the angle at where they meet, is the right angle