Question

Janet draws triangle PQR with vertices P (1, 1), Q (-2, 4), and R (1, 7). She claims that the triangle is an isosceles right triangle.

Answer 1

Check the side lengths and angle at Q.

Vectors and squared lengths:
- PQ = Q − P = (−2−1, 4−1) = (−3, 3) ⇒ |PQ|^2 = 9+9 = 18.
- QR = R − Q = (1−(−2), 7−4) = (3, 3) ⇒ |QR|^2 = 9+9 = 18.
- PR = R − P = (1−1, 7−1) = (0, 6) ⇒ |PR|^2 = 36.

So PQ = QR (both √18), so the triangle is isosceles with equal sides meeting at Q.

Check perpendicularity at Q (dot product of QP and QR):
- QP = P − Q = (3, −3), QR = (3, 3); QP · QR = 3·3 + (−3)·3 = 9 − 9 = 0.

Since the two equal sides are perpendicular, the triangle is an isosceles right triangle (right angle at Q). Janet is correct.