Official Solution: If vertices of a triangle have coordinates \((-1, 1)\), \((4, 1)\), and \((x, y)\), what is the area of the triangle? (1) \(y^2-2y-3=0\).
Solving this equation, we get \(y=3\) or \(y=-1\). Consider the diagram below:
The third vertex is either on the line \(y=3\) (blue line) or on the line \(y=-1\) (green line). Observe that in either case, the height of the triangle is 2 (based on the two possible positions of the third vertex shown in the diagram). Since the length of the base of the triangle is \(4-(-1)=5\), the area can be calculated as \(\frac{1}{2}*base *height = \frac{1}{2}*5*2 = 5\). Sufficient.
(2) \(x^2=y^2\).
This implies that \(|x|=|y|\). However, this statement alone is clearly insufficient to determine the area of the triangle, as various possibilities for \((x, y)\) exist, such as \((x, y) = (3, 3)\) or \((x, y) = (5, 5)\).
Answer: A