In the rectangle coordinate system, triangle ABC has a

(1) The third vertex of triangle ABC lies on the x-axis, and the triangle has an area of 196 --> ABC is a right triangle --> \(\frac{|x|*56}{2}=196\), where \(x\) is the x-coordinate of the third vertex --> \(|x|=7\) --> the third vertex is at the point (7, 0) OR (-7, 0).

I think there is a little problem with the question stem: the line passing through the points A and C is either \(y+8x=56\), if the third vertex is at (7, 0) or \(y-8x=56\), if the third vertex is at (-7, 0). Now, there are infinitely many integer pairs of (x,y) satisfying each equation but on the GMAT for the question which asks for a certain value of an unknown, a statement is sufficient if it gives single numerical value of this unknown. So I think the question should say "how many points on line segment AC have integer values for both their x and y values". And in this case statement (1) would be sufficient.

(2) Point A has a positive x coordinate and a y coordinate of zero. Clearly insufficient.