# Problem of the Day

In the -coordinate plane, is the point further from the point than it is from the point ?

(1) (2) 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 joiningÂ \$(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?