Let S be the set of all triangles in the xy -plane, each having one...

Since the vertex of given triangle is $O$ (origin) and other two vertices A and B$lies on the coordinate axes, hence triangle must be a right-angled triangle.<br /><br />Area = \dfrac{1}{2}x$. y ( x, y \in I )$<br /><br />As per given condition<br /><br />\dfrac{1}{2}x$. y = 50 ⇒ x . y = 100

Possible ordered pairs of \left(x, y\right) when$∆$ formed in ${1}^{st}$ quadrant are $9$ i. e. \left(1, 100\right)$, \left(2, 50\right)$, \left(4, 25\right)$, \left(5, 20\right)$, \left(10, 10\right), \left(25, 4\right), \left(50, 2\right), \left(100, 1\right)

$∆$ can be formed in any of the quadrants, hence total number of cases 9 \times 4 = 36 .