Consider the set of all triangles OPQ where O | vedclass

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(C) The coordinates of $P$ and $Q$ satisfy $y = 99 - 5x$. Since $x, y \ge 0$,$5x \le 99$,so $x \in \{0, 1, 2, \dots, 19\}$. There are $20$ such points

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$90$

Vedclass · Answer

(C) The coordinates of $P$ and $Q$ satisfy $y = 99 - 5x$. Since $x, y \ge 0$,$5x \le 99$,so $x \in \{0, 1, 2, \dots, 19\}$. There are $20$ such points.Let $P = (x_1, 99 - 5x_1)$ and $Q = (x_2, 99 - 5x_2)$.The area of $\Delta OPQ$ is given by $\Delta = \frac{1}{2} |x_1(99 - 5x_2) - x_2(99 - 5x_1)|$.$\Delta = \frac{1}{2} |99x_1 - 5x_1x_2 - 99x_2 + 5x_1x_2| = \frac{99}{2} |x_1 - x_2|$.For the area to be an integer,$|x_1 - x_2|$ must be even.This means $x_1$ and $x_2$ must have the same parity (both even or both odd).There are $10$ even values $(0, 2, \dots, 18)$ and $10$ odd values $(1, 3, \dots, 19)$.The number of ways to choose two distinct points is $\binom{10}{2} + \binom{10}{2} = 45 + 45 = 90$.

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$,$Q$ are distinct points in the plane with non-negative integral coordinates $(x, y)$ such that $5x + y = 99$. The number of such distinct triangles whose area is a positive integer is:

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