I could suggest a different approach here. Notice that OP and OQ are perpendicular lines, and are the same length. We can use the simple fact that they are perpendicular to get the answer directly here, with no calculations.
To review the theory (and I'll use concrete numbers to make things easier): if 2/3 is the slope of a line L, we know that -3/2 is the slope of a perpendicular line M. Now, what does the slope '2/3' mean? It means you need to go right 3 units in order that L will rise by 2 units. That's the definition of the slope. On the perpendicular line M, with slope -3/2, you would need to go right 2 units in order for the line to 'rise' -3 units (i.e. fall by 3 units). The vertical and horizontal changes get 'reversed' on the perpendicular line, and one of the changes becomes negative. Thus, if (0,0) is a point on both L and M, then we know that (3, 2) will be a point on L, and (2, -3) will be a point on the perpendicular line M. Notice that each of these points is the same distance from (0,0).
The relationship between perpendicular lines becomes much clearer if you draw a few examples on the co-ordinate plane. Anyway, if you understand this relationship well, the question can be solved almost immediately:
On OP, if we go right by sqrt(3) units, the line falls by 1 unit. On the perpendicular line OQ, if we go right by 1 unit, the line must rise by sqrt(3) units, since OQ is perpendicular to OP. Thus, point Q must be (1, sqrt(3)).
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