Question

PLEASE HELP! MEDAL WILL BE AWARDED!

The vertices of a quadrilateral, OABC, are (0,0), (4,2), (6,10) and (2,8) respectively. Use a vector method to answer the questions which follow.

a. (i) State two geometrical relationships between the line segments OA and CB.
(ii) Explain why OABC is a parallelogram

b. If M is the midpoint of the diagonal OB, and N is the midpoint of the diagonal AC, determine the position vector of:
(i) OM
(ii) ON

Hence, state one conclusion which can be made about the diagonals of the parallelogram.

Answer 1

for a, OA and CB are pararell and has the same length. Because their "directional" vector is (6,10)-(2,8)=4,2 and (4,2)-(0,0)=4,2 and the length is from SQRT((6-2)^2+(10-8)^2)=Sqrt(20) and SQRT((4-0)^2+(2-0)^2)=Sqrt(20). Other sides are also paralell so its a paralellogramm

for b, OM=OB/2=(6,10)/2=(3,5), same for ON.
So the statement is that OM and ON are equal. The midpoints are in the same place.