Question

Points $A = (3, 9),$ $B = (1, 1),$ $C = (5, 3),$ and $D = (a, b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD.$ The quadrilateral formed by joining the midpoints of $\overline{AB},$ $\overline{BC},$ $\overline{CD},$ and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D?$

$7$

$9$

$10$

$12$

$16$

Solution:

The midpoints are $M = (2, 5)$ of $\overline{AB}$ and $N = (3, 2)$ of $\overline{BC}.$

For the midpoint quadrilateral to be a square, consecutive sides are perpendicular and equal. With $\overrightarrow{NM} = \langle -1, 3 \rangle,$ the side $\overrightarrow{MQ}$ to the midpoint $Q$ of $\overline{DA}$ must be $\langle 3, 1 \rangle,$ so $Q = (5, 6).$

Since $Q$ is the midpoint of $\overline{DA}$ and $A = (3, 9),$ we get $D = 2Q - A = (7, 3).$ The sum of its coordinates is $7 + 3 = 10.$

Thus, the correct answer is C.

2001 AMC 12 Problem 20

Solution:

Problem 20 in Other Years