Question

Show all caclulations for credit! The coordinates of the vertices of triangle ABC are A(-1,3), B(1,2) and C(-3,-1). Determine the slope of each side of the triangle and use that information to determine if the triangle is a right triangle or not.

Answer 1

Slope (m) of a line = (y2-y1)/(x2-x1)
So for line AB, the slope is (3-2)/(-1-1)
= 1/-2 = -1/2
For line AC = (3--1)/(-1--3) = (3+1)/(3-1)
= 4/2 = 2
For line BC = (2--1)/(1--3) = (2+1)/(1+3)
= 3/4
In order for an angle (<) to be right, it must be 90°, so the two lines making the right angle must be perpendicular. Perpendicular lines by definition have slopes that are the negative reciprocal. That means that you change the sign of one line's slope (m) and divide 1 by it:
m2 = 1/-m1
So for lines AB and AC: m(AC) = 1/-m(AB),
does 2 = 1/--1/2? YES!! 1/--1/2 = 1/1/2 = 2, so < BAC is 90° and therefore a right <
How about for lines AB and BC: m(BC) = 1/-m(AB), does 3/4 = 1/--1/2? NO, because 1/--1/2 = 1/1/2 = 2, not = to 3/4, so < ABC is not right
How about our last < BCA: m(AC) = 1/-m(BC), does 2 = 1/-3/4? NO, because 1/-3/4 = -4/3, not = to 2, so < BCA is not right
So yes, the triangle is a right triangle because < BAC is right (=90°)!