Problem:
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 AB,BC,CD, and DA is a square. What is the sum of the coordinates of point D ?
Answer Choices:
A. 7
B. 9
C. 10
D. 12
E. 16 Solution:
Let the midpoints of sides AB,BC,CD, and DA be denoted M,N,P, and Q, respectively. Then M=(2,5) and N=(3,2). Since MN has slope -3 , the slope of MQ​ must be 1/3, and MQ=MN=10​. An equation for the line containing MQ​ is thus y−5=31​(x−2), or y=(x+13)/3. So Q has coordinates of the form (a,31​(a+13)). Since MQ=10​, we have
\section*{OR}
Each pair of opposite sides of the square are parallel to a diagonal of ABCD, so the diagonals of ABCD are perpendicular. Similarly, each pair of opposite sides of the square has length half that of a diagonal, so the diagonals of ABCD are congruent. Since the slope of AC is -3 and AC is perpendicular to BD, we have