How do you determine whether a triangle is a right triangle, given the coordinate points of its vertices?

Question

Sample Problem:

Triangle GHI

G (2,7) H (3,6) I (-4,-1)

Answer 1

if you find the slopes of any two sides to be negative reciprocals of one another, that proves side perpendicularity (and therefore a rt angle).

 

if you calculate the lengths of all three sides, and call them a, b & c, with c being the greatest of these values, and if these satisfy the Pythagorean Theorem (c sauared = sum of squares of a & b), then it's a right triangle.

Answer 2

In a right triangle two of the sides have to be perpendicular to each other. If the slopes of the sides are negative reciprocals of each other, then the sides are perpendicular.

m = y₂ - y₁

        x₂ - x₁

m_HI = (6 + 1)/(3 + 4) = 7/7 

m_HI = 1

 

m_GH = (7-6)(2-3) = 1/-1

m_GH = -1

 

Since the slopes are negative reciprocals of each other, HI⊥GH.

Thus, the triangle is a right triangle.