2001 AMC 12 Problem 20

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

Let

M

and

N

be the midpoints of

\overline{AB}

and

\overline{BC};

find

M

and

N

first

The next midpoint

Q

satisfies

\overrightarrow{MQ} = \overrightarrow{NM}

rotated

90^\circ

(same length, perpendicular)

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.

Problem 20 in other years