In the rectangular coordinate system above, if AB < BC, is the area of

Official Explanation

The question asks whether the area of the triangle is greater than 12. For the area of the triangle, we will most likely be looking for a valid base-height pair. AC would be a valid base, and in that case the x coordinate of point B would give us the dimension of a valid height. Let's turn to the data statements, evaluating them separately first, as always.

Statement (1) tells us the coordinates of point B. We have the height of ABC, but not the base. But one thing to keep in mind in coordinate geometry is that every point location describes a right triangle. So here, we can actually find the area of the little triangle defined by points A, B, and (0, -3). It has a valid base/height pair of 3 and 4 so its area is 6. Moreover, since AB < BC, this sub-triangle is smaller than the other triangle that is part of ABC, so the area of that part is greater than 6. Therefore the area of ABC must be greater than 12. Statement (1) is sufficient.

Statement (2) is similar in that we have the base of ABC, but no height. Can we make a tricky inference here as we did for Statement (1)? We cannot. Statement (1) gave us two dimensions of information about the triangle, but there is only one dimension of information in the point that lies on the y-axis. Point B could be close to the axis, making a small triangle, or far out, making a large one. Statement (2) is insufficient.

The correct answer is (A).

Video Explanation