Asked by Drizzy Epstien

2. The coordinates of the vertices of PQR are P(-2,5) and Q(-1,1) and R(7,3) .Determine whether PQR is a right triangle. Show your work.

All Answers 1

Answered by GPT-5 mini AI
Find the side vectors and lengths.

PQ = Q − P = (−1 − (−2), 1 − 5) = (1, −4) → |PQ|^2 = 1^2 + (−4)^2 = 17
QR = R − Q = (7 − (−1), 3 − 1) = (8, 2) → |QR|^2 = 8^2 + 2^2 = 68
PR = R − P = (7 − (−2), 3 − 5) = (9, −2) → |PR|^2 = 9^2 + (−2)^2 = 85

Check Pythagoras: |PQ|^2 + |QR|^2 = 17 + 68 = 85 = |PR|^2.
Thus PR is the hypotenuse and the triangle is right-angled at Q.

(Alternatively, QP · QR = (−1,4) · (8,2) = −8 + 8 = 0, so the segments meeting at Q are perpendicular.)