A line segment is drawn in the xy-coordinate plane with endpoints P(3,

Given that a line is drawn from P(3,5) to Q(8,0). We know that line n intersects PQ, but exactly where line n intersects line PQ, we don't know. We are to determine the slope of line n.

Statement 1: Point A is located on line n, and AP=AQ
Statement 1 is insufficient. This is because from statement 1, line n can be a locus of points equidistant from points P and Q. In this case, line n is a perpendicular line to line PQ passing through the midpoint between the endpoints on line PQ, and the slope will be the negative reciprocal of the slope of line PQ.
But if by definition of point A on line n whereby AP=AQ implies that point A is the point of intersection between line n and line PQ, then we cannot conclusively determine a specific slope for line n. since n doesn't necessarily have to be perpendicular to line PQ for it to intersect line PQ at A such that AP=AQ.

Statement 2: The x-intercept of line n is (3,0).
Statement 2 is also insufficient. This is because we don't know the specific point where line n and line PQ intersect. So we cannot determine a specific slope for line n based on statement 2 alone.

1+2
Insufficient. Earlier on, I was a proponent for the view that both statements when taken together is sufficient. However, I had to change my mind on the sufficiency of combining 1 and 2. Why? If this question was well thought out as I believe it indeed was, statements 1 and 2 combined will still not be sufficient. We need to understand that the x-intercept of n (3,0) is also equidistant from P and Q.

To verify, let B denote the x-intercept of n. So B(3,0). Distance between P(3,5) and B(3,0)=√(0^2+5^2)=5.
Distance between B(3,0) and Q(8,0) = √((8-3)^2+0)=5.
What about if point A mentioned is actually the same point as the x-intercept? This is possible since the x-intercept of n is also equidistant from P and Q. Statement 1 is not definite on the specific location of point A on line n. So, A can actually be the x-intercept as given in statement 2. When this happens, then we cannot definitely say that n is perpendicular to PQ. As a result, statements 1 and 2 even when taken together are not sufficient.

The answer is, therefore E.