Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and line PR is parallel to the x...

Question

Right triangle PQR is to be constructed in the xy-plane so that the right angle is at P and line PR is parallel to the x-axis. The x and y coordinates of P, Q, and R are to be integers that satisfy the inequalities: -4 <= x <= 5 and 6 <= y <= 16. How many different triangles could be constructed with these properties?

A. 110
B. 1,100
C. 9,900
D. 10,000

Accepted Answer

9,900 distinct right triangles can be constructed. We need to count integer-coordinate triangles where the right angle is at P and side PR is horizontal (parallel to x-axis). Since PR is horizontal, P and R share the same y-coordinate. Since the right angle is at P and PR is horizontal, PQ must be vertical (perpendicular to PR), so P and Q share the same x-coordinate. This reduces the problem to independently choosing x-coordinates and y-coordinates.