Question

how do i use the triangular inequality theorm to prove that the points (-5,5) , (4, -6) and (15, -6) form a triangle?

Answer 1

Hmm.. Well the triangle inequality is about distances and side lengths.
If you are given three line segments a, b, and c such that
a + b > c, a + c > b, and b + c > a, then
these side lengths can be used to form a triangle.

I don’t particularly see the point in doing that for this problem,
since that would require calculating the distances from each point to the next.
Then we’d have to check a + b > c, a + c > b, and b + c > a.

But that’s really unnecessary.
To show that the points form a triangle, all you need to do is to show that the three points
don’t all lie on the same line.
This is because if you put three random points on the plane,
they will always form a triangle unless they are all on the same line.

We see that the last two points lie on the line y = -6.
But the first one doesn’t lie on that line.
So the three points aren’t all collinear, and so they form a triangle.