# Problem of the Day

A line segment is drawn from point on triangle to point , the midpoint of side . If is a right angle, What is the length of ?

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

See the Solution

### Solution

[latexpage] This is what the triangle looks like based on the information before the statements. It turns out that a median drawn from the right angle of a right triangle to the hypotenuse is always exactly one half the length of the hypotenuse. That is $AD=BD=CD$. To see why, remember that any right triangle can be inscribed in a circle such that the hypotenuse is the diameter. If we do that, $\overline{AD}$, $\overline{BD}$, and $\overline{CD}$ are all radii of the circle and are all therefore congruent to each other. So, if we know the lengths of either $\overline{BD}$ or $\overline{CD}$ we will know the length of $\overline{AD}$.

Statement 1: SUFFICIENT. If $BD=5$ then $AD=5$.

Statement 2: INSUFFICIENT. Remember that $AD$ is always equal to one-half the length of the hypotenuse. If $AB=6$, the length of the hypotenuse will change if we pick different values of $AC$.