Problem of the Day


In the xy-coordinate plane, is the point (a,b) further from the point (2,3) than it is from the point (3,2)?

(1) a=2.6

(2) b>a

Reveal Answer

Answer

B. Statement 2 alone is sufficient, but statement 1 alone is not sufficient.

See the Solution

Solution

[latexpage]

First, note that all of the points that are equidistant from the points $(2,3)$ and $(3,2)$ lie on the perpendicular bisector of the line segment joining $(2,3)$ and $(3,2)$. To find the equation of this line, note that the midpoint of this line segment is $(2.5,2.5)$ and the slope is $-1$. Thus, the slope of the perpendicular bisector must be $1$. From this point and slope, we can deduce that the equation of the perpendicular bisector is $y=x$. Any points that lie above this line are closer to $(2,3)$ and any points that lie below this line are closer to $(3,2)$.

(1) $a=2.6$ only tells us the $x$-coordinate of $(a,b)$. Without the $y$-coordinate, we can’t tell which side of the line $y=x$ it is on. INSUFFICIENT.

(2) $b>a$ tells us that the $y$-coordinate is bigger than the $x$-coordinate. This means the point must lie above the line $y=x$, and thus must be closer to $(2,3)$. SUFFICIENT.


One Response

  1. David says:

    this problem sucks. where the hell are the answer choices?