Problem: Three vertices of parallelogram PQRS are P(−3,−2),Q(1,−5),R(9,1) with P and R diagonally opposite. The sum of the coordinates of vertex S is:
Answer Choices:
A. 13
B. 12
C. 11
D. 10
E. 9
Solution:
The sum of the x-coordinates of P and R equals the sum of the x-coordinates of Q and S, and, similarly, for the y-coordinates. ∴x+1=6,x=5 and y−5=−1,y=4∴x+y=9
or
Let Sx​ be the x-coordinate of S and Sy​, the y-coordinate of S, and, similarly, for P,Q, and R. Then, by congruent triangles. Qx​−Rx​=Px​−Sx​,1−9=−3−Sx​,Sx​=5. Similarly Sy​=4. ∴Sx​+Sy​=9.