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 in coordinate axes with integral coordinates. If each triangle in S has area of 50 sq. units then the number of elements in the set S is
(a) 9
(b) 18
(c) 32
(d) 36

Answer 1

Hint: We will first find the area of the given triangle and then equate it to the area given to us in the problem. This would give us a condition that contains the variables, height and the base of the triangle. Once we solve that condition we would get the different pairs of height and base that is in turn the number of coordinates. The number of pairs of coordinates would give us the distinct number of triangles.

Complete step-by-step solution -
Let us now begin with the solution.
To solve this problem we just need the basic formula for area and some idea about factoring numbers. The factors representation is also important as it might result in more or less number of answers.
Area of the triangle \[=\dfrac{1}{2}bh\]
Now as it is given that one of the coordinates lie on the X-axis, let that coordinate be (a1, 0).
Also it is given that the other coordinate lies on the Y-axis. Let that be (0, a2)
Now the distance from the origin to these points is nothing but the length of that side.
Thus one of the lengths is a1 and the other length is a2 .
Therefore,
Area of the triangle \[=\dfrac{1}{2}bh\]
\[50=\dfrac{1}{2}{{a}_{1}}{{a}_{2}}\]
Cross-multiplying we get,
100 = \[{{a}_{1}}{{a}_{2}}\]
Now we need to find a1 and a2.
Now let us find the pairs of numbers whose product result in 100.
Therefore,
\[\begin{align}
  & 100=1\times 100 \\
 & 100=2\times 50 \\
 & 100=4\times 25 \\
 & 100=5\times 20 \\
 & 100=10\times 10 \\
 & 100=20\times 5 \\
 & 100=25\times 4 \\
 & 100=50\times 2 \\
 & 100=100\times 1 \\
\end{align}\]
Thus the pairs are, (1,100), (2,50), (4,25), (5,20), (10,10), (20,5), (25,4), (50,2) and (100,1).
Thus there are 9 pairs.
So there are 9 pairs of coordinates.
Thus there are 9 triangles in the set S.
Thus the correct option is option(a)

Note: You might get confused as to why to take 2 and 50 two times differently or 1 and 100 two times differently. The reason is that though they commute in the terms of multiplication they do not commute as co-ordinates and they would form different points in different arrangements. Thus there are 9 different points.