Question

Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area $50$ sq. units, then the number of elements in the set $S$ is :

• A. $9$
• B. $18$
• C. $36$
• D. $32$

Answer 1

The correct option is C $36$


Area $=\frac{1}{2}\times |a|\times |b|=50$
$\Rightarrow |a|\cdot |b|=100$
So the possible pairs $(|a|,|b|)$ are $(1,100),(100,1),(2,50),(50,2),(4,25),(25,4),(5,20),(20,5),(10,10)$
No. of possible pairs $=9$

As we know, one pair can draw $4$ triangles,
so the total number of possible triangles $=9\times 4=36$