Let S be the set of all triangles in the xy-plane, each having one vertex at the origin an…

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:

Accepted Answer

Vertices $A(a,0)$, $B(0,b)$, with area $=\dfrac12|ab|=50 \implies |ab|=100$.

Factor pairs of 100 (unordered, positive): $1\times100, 2\times50, 4\times25, 5\times20, 10\times10$.

For factor pairs where the two factors differ ($1\times100,2\times50,4\times25,5\times20$): each gives $4$ sign combinations $\times 2$ (order of $a,b$) $= 8$ triangles; there are 4 such pairs: $4\times8=32$.

For $10\times10$ (equal factors): $4$ sign combinations only (order doesn't create new triangles): $4$.

$$\text{Total} = 32+4 = 36$$

Solutions · 1