Question

In the xy-coordinate plane, is point R equidistant from points (-3,-3) and (1,-3) ?
(1) The x-coordinate of point R is -1.
(2) Point R lies on the line y = -3.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is A Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

• We have to determine whether point R is equidistant from (–3,−3) and (1,−3). Letting R = (x,y), then R will be equidistant from (–3,−3) and (1,−3) if and only if R lies on the perpendicular bisector of the line segment with endpoints (−3,−3) and (1,−3), or equivalently, if and only if R lies on the vertical line that consists of all points with x-coordinate equal to . Therefore, determine if x = −1.
  

1. From statement 1, it is given that x = −1, then R will be equidistant from (−3,−3) and (1,−3); This is sufficient.
2. From statement 2, it is given that y = −3, then both x = −1 and x ≠ −1 are possible; This is not sufficient.

• Alternatively, letting R = (x,y), then R will be equidistant from (−3,−3) and (1,−3) if and only if the distance between (x,y) and (−3,−3) is the same as the distance between (x,y) and (1,−3), or if and only if or if and only if or if and only if .
• Given that x = −1, then and ; SUFFICIENT.
• Given that y = −3, then it is impossible to determine if ; NOT sufficient.
• The correct answer is A; statement 1 alone is sufficient.