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Let S be the set of all triangles in the XY-plane, each with one vertex at the origin and the other two vertices on the coordinate axes at integer coordinates. If each triangle in S has area 50 sq. units, find the number of elements in S.
- 9
- 18
- 32
- 36
Correct answer: 36
Solution
Area = (1/2)|a||b| = 50, so |a||b| = 100. Number of positive divisor pairs of 100 = 9 (since 100 = 2² * 5² has (2+1)(2+1) = 9 divisors). Each pair (|a|,|b|) gives 4 sign combinations. Total = 9 * 4 = 36.
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