The vertices of a triangle are located at the points: (-2, 1), (3, 1), (x, y) on a coordinate plane. What is the area of the triangle?
(1) |y – 1| = 1
(2) The interior angle at vertex (x, y) measures 90 degrees.
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(A) From Statement (1), we can determine that y is either 2 or 0. In either case, the height of the triangle is 1 (the distance from 1 to either 2 or 0 is 1), and the base of the triangle is 5 (the distance between -2 and 3), so we can calculate the area of the triangle (we know that the height is 1 and the base is 5). Therefore, Statement (1) is sufficient to calculate the area of the triangle.
Note: to calculate the area of a triangle, we just need to know the lengths of the base and height. The shape of the triangle does not matter. That is why we do not care what the value of x is and don’t need it for Statement (1) to be sufficient.

Now let’s evaluate Statement (2). Though we know the triangle is a right triangle with a hypotenuse of length 5, we cannot determine the length of the legs of the triangle. So there are infinitely many possibilities for the area of the triangle. For example, it could have legs of length 3 and 4 (making the area 6) or it could have legs that are both 5/√2 (making the area 25/4). Therefore, Statement (2) is insufficient.

Since Statement (1) is sufficient alone and Statement (2) is insufficient, the correct answer is choice (A).
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For point (1) to be sufficient, we need to know that x and y can only be integers.

No, x can poses any value.

For any x and y = 0 or y = 2 the height length to the base (between points (-2, 1) and (3, 1)) is the same.

Let's consider the case y = 2 (y = 0 has the same reasoning).
One side of the triangle (between points (-2, 1) and (3, 1)) is set. The point (x, y) lies somewhere on the line y = 2:


Any such triangle has the same base and the length of the height to the base:

Based on statement (1), the area of the triangle is 2.5 . We have proven it in the explanation.

As you can see on the following image, the triangles defined by statement (1) are not necessarily right:



In fact, statement (2) defines such triangles, that the vertex (x, y) lies on the circle with the segment (-2, 1) (3, 1) as the diameter.