Given the coordinates of the vertices of a triangle, what is an efficient way of classifying said triangle?

Question

Given the coordinates of all vertices of a triangle, what is an efficient way of determining which of the followig propreties the triangle has?

i) Right

ii) Isoceles

iii) Scalene

I am asking this question in order to be able to help college algebra students so I am interested in an approach using only basic tools, but I am also more advanced tools for my own personal interest.

I know that ii) and iii) can be determined by just calculating the length of each side using the distance formula. i) can be calculated by checking if the slopes of any of the sides are negative reciprocals of each other. But this is very time consuming; is there a better way?

Answer 1

I don't know that you can do much better than actually calculating the $3$ squared lengths of the sides (squared, as to save some square-root calls) then:

• if either are equal, then it's an isosceles triangle;

• if the largest is the sum of the other two then it's a right triangle;

• if neither of the above (which are not mutually exclusive), then it's a scalene triangle.