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Vector Algebra

Question
CBSEENMA12034040
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The mid-points of two opposite sides of a quadrilateral and the mid-points of the diagonals are the vertices of a parallelogram. Prove using vectors.

Solution

Let straight a with rightwards arrow on top comma space straight b with rightwards arrow on top comma space straight c with rightwards arrow on top comma space straight d with rightwards arrow on top be position vectors of vertices A, B, C, D repsectively.
Let E, F, G, H be mid-points of AB, CD, AC, BD respectively.
P.V. of E = fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction
P.V. of F = fraction numerator straight c with rightwards arrow on top plus straight d with rightwards arrow on top over denominator 2 end fraction
P.V. of G = fraction numerator straight a with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 2 end fraction
P.V. of H = fraction numerator straight b with rightwards arrow on top space plus straight d with rightwards arrow on top over denominator 2 end fraction

EG with rightwards arrow on top space equals space straight P. straight V. of space straight G space minus space straight P. straight V. space of space straight E space equals space fraction numerator straight a with rightwards arrow on top plus straight c with rightwards arrow on top over denominator 2 end fraction minus fraction numerator straight a with rightwards arrow on top plus straight b with rightwards arrow on top over denominator 2 end fraction space equals fraction numerator straight c with rightwards arrow on top minus straight b with rightwards arrow on top over denominator 2 end fraction HF with rightwards arrow on top space equals space straight P. straight V. space of space straight F space minus space straight P. straight V. space of space straight H space equals space fraction numerator straight c with rightwards arrow on top plus straight d with rightwards arrow on top over denominator 2 end fraction minus fraction numerator straight b with rightwards arrow on top plus straight d with rightwards arrow on top over denominator 2 end fraction space equals fraction numerator straight c with rightwards arrow on top minus straight b with rightwards arrow on top over denominator 2 end fraction
therefore space space space space EG with rightwards arrow on top space equals space HF with rightwards arrow on top space space space space space rightwards double arrow space space space space EG thin space vertical line vertical line thin space HF space and space EG space equals space HF space space space space rightwards double arrow space space space EGHF space is space straight a space parallelogram. space

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