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GCSE Math Vectors Question?

• GCSE Math Vectors Question: ABCD is a trapezium with BC Parallel to AD AB = 2x BC= 2y AD= 6y N= Midpoint of BC M= Midpoint AD P = Midpoint NM Q= Midpoint CD (Help with labelling: Bottom Lefy= A, Top Left= B, Top Right = C, Bottom Right = D) Based on my basic knowledge PQ should = 3y Workings: PN = x NC = y CQ= 1/2 CD (since its the midpoint) So find CD = -2y + - x + 6y = 4y + - x So... PN + NC + CQ = x + y + 1/2 (4y-x) = x + y + 2y - x should = 3y But the revision guide i'm using claims it is 2y My question was to prove PQ is Parallel with AD because PQ (3y) is a multiple of AD (6y) it is therefore parallel; however although it would still be parallel, it claims for some reason that vector PQ is 2y, is there a error in my calculations to think it should be 3y since i'm currently revising ready for my GCSE's but this question seems to confuse me. Thanks for reading, all help will be very much appreciated :D

• Answer:

  If you look at a sketch of the figure, you have NC = y MD = 3y Why would it make sense for PQ = 3y? Does it seem reasonable for PQ to be the same length as MD? PQ should be the average of NC and MD or PQ = (NC + MD)/2 = (y + 3y)/2 PQ = 2y