Discussion
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: \(-4 \le x \le 5\) and \(6 \le y \le 16\). How many different triangles could be constructed with these properties?
Answer: C
9,900 triangles can be constructed.
The x-coordinate can take 10 integer values (\(-4\) to 5) and the y-coordinate 11 values (6 to 16).
- P can be any of the \(10 \times 11 = 110\) lattice points.
- PR is horizontal, so R has the same y as P and a different x: 9 choices.
- The right angle is at P, so PQ is vertical: Q has the same x as P and a different y: 10 choices.
\[ 110 \times 9 \times 10 = 9900 \]
110 counts only the choices of P, and 1,100 forgets one of the factors; 10,000 uses 10 instead of 9 for R, allowing R to coincide with P.
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