# Problem of the Day

In the figure above, point lies on a horizontal line that is parallel to the -axis. Â What is the area of ?

(1)
(2)

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

See the Solution

### Solution

[latexpage]

To find the area of a triangle, we can use the formula $A=\frac{1}{2} b \cdot h$. Â From the figure, we can deduce that the base, $\overline{AB}$ has a fixed length of $3$. Â So, the area will be determined by any statement that fixes the height. Â The height is the length of the perpendicular segment drawn from $C$ to the line containing the base, $\overline{AB}$.

(1) SUFFICIENT. Â This statement determines the height of the triangle. Â If the $y$ coordinate of $C$ is $4$, then the $y$ coordinate of every point on the line is $4$. Â We know this because we are told it is a horizontal line that is parallel to the $x$-axis. Therefore, the height of the triangle will always be $4$, no matter what the $x$-coordinate of $C$ is.

(2) INSUFFICIENT. This statement gives us no information about the height of the triangle.

Notice that once the height is fixed, the value of the $x$-coordinate becomes irrelevant to the problem of finding the area. Â We could slide point $C$ $1,000,000$ miles down that line and the triangle would still have the same area. Â Other aspects of the triangle, such as the perimeter, would change, but the area remains constant.