Question

The coordinates of the vertices of Δpqr are p(2,2), q(7,1), and r(4,-1). Using slopes (not the Pythagorean theorem), determine whether Δpqr is a right triangle. Show your work.

1) Yes, Δpqr is a right triangle
2) No, Δpqr is not a right triangle
3) Cannot be determined

Answer 1

ΔPQR is not a right triangle because the slopes of the lines PQ and PR, when multiplied, do not equal -1, which would indicate perpendicular lines.

Step-by-step explanation:

To determine if ΔPQR is a right triangle using the slopes of its sides, we need to calculate the slopes of two lines and see if they are perpendicular. Perpendicular lines have slopes that are negative reciprocals of each other.

First, calculate the slope of line PQ. The slope m is the change in y divided by the change in x: mPQ = (1 - 2) / (7 - 2) = -1 / 5. Next, calculate the slope of line PR: mPR = (-1 - 2) / (4 - 2) = -3 / 2

If ΔPQR is a right triangle, the product of these two slopes should be -1. Let's check: mPQ × mPR = (-1 / 5) × (-3 / 2) = 3 / 10. Since the product of the slopes of PQ and PR is not -1, they are not perpendicular, and ΔPQR is not a right triangle.