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). ---------- For point (1) to be sufficient, we need to know that x and y can only be integers.
What if we consider another side of the triangle as base instead the one with the length five? As in the answer is A if we consider the base with length 5, but it falls apart if we consider some other side as a base? We cannot calculate the area then, can we?
If we consider another side of the triangle as the base, then we will consider another segment as the height. However the area of such triangle will be the same, as we calculated it using the segment [(-2, 1), (3, 1)] as the base. The area is a property of a triangle and thus it will be the same regardless of the way we calculate it.
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