Question

2. The coordinates of the vertices of are , , and . Determine whether is a right triangle. Show your work.

Answer 1

It looks like the coordinates were left out of your message. Please paste the three vertex coordinates (for example, A(x1,y1), B(x2,y2), C(x3,y3)) and I will check whether the triangle is a right triangle and show the work.

How I'll do it (two equivalent methods — you can pick one):

Method 1 — distance / Pythagorean check
1. Compute the squared lengths of the three sides:
AB^2 = (x1−x2)^2 + (y1−y2)^2
BC^2 = (x2−x3)^2 + (y2−y3)^2
CA^2 = (x3−x1)^2 + (y3−y1)^2
2. Sort the three squared lengths so the largest is on one side. If the sum of the two smaller squared lengths equals the largest squared length, the triangle is right-angled (Pythagorean theorem).

Method 2 — slope / perpendicular check
1. Compute slopes of two sides that meet at a vertex, e.g. slope AB = (y2−y1)/(x2−x1) and slope AC = (y3−y1)/(x3−x1) (watch for vertical lines).
2. If the product of the slopes of any two sides that share a vertex is −1, those sides are perpendicular and the triangle is right.

Send the actual coordinates and I will carry out the calculations step by step.