If P and Q are the mid-points of the sides AB and CD of a parall

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 Multiple Choice QuestionsLong Answer Type

121. Prove, using vectors, that the line segment joining the mid-points of the non-parallel sides of a trapezium is parallel to the bases and is equal to half the sum of their lenghts.
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122. 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.
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123. If D, E and F are the mid-points of the sides of a triangle ABC, show that OA with rightwards arrow on top space plus space OB with rightwards arrow on top space plus space OC with rightwards arrow on top space equals space OD with rightwards arrow on top space plus space OE with rightwards arrow on top space plus space OF with rightwards arrow on top, where O is any arbitrary point.
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124. The points D, E, F divide the sides BC, CA. AB of a triangle in the ratio 1 : 4, 3 : 2 and 3 : 7 respectively. Show that the sum of the vectors AB with rightwards arrow on top comma space BE with rightwards arrow on top space and space CF with rightwards arrow on top is parallel to CK with rightwards arrow on top comma space where K divides AB in the ratio 1 : 3.
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125. Show that the line joining one vertex of a parallelogram to the mid-point of an opposite side trisects the diagonal and is trisected there at.
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126. If P and Q are the mid-points of the sides AB and CD of a parallelogram ABCD, prove that DP and BQ cut the diagonal AC in its points of trisection which are also the points of trisection of DP and BQ respectively.


Take A as origin. Let straight b with rightwards arrow on top space and space straight d with rightwards arrow on top be position vectors of B and D respectively such that
AB with rightwards arrow on top space equals space straight b with rightwards arrow on top comma space space AD with rightwards arrow on top space equals space straight d with rightwards arrow on top

Now position vector of P, mid-point of AB, is fraction numerator straight d with rightwards arrow on top over denominator 2 end fraction.
Also,    AQ with rightwards arrow on top space equals space AD with rightwards arrow on top space plus space DQ with rightwards arrow on top space equals space AD with rightwards arrow on top space space plus space 1 half DC with rightwards arrow on top space equals AD with rightwards arrow on top space plus space 1 half AB with rightwards arrow on top space equals space straight d with rightwards arrow on top space plus space 1 half straight b with rightwards arrow on top
therefore  position vector of Q is straight d with rightwards arrow on top space plus space fraction numerator straight b with rightwards arrow on top over denominator 2 end fraction
              AC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space BC with rightwards arrow on top space equals space AB with rightwards arrow on top space plus space AD with rightwards arrow on top space equals space straight b with rightwards arrow on top plus straight d with rightwards arrow on top
Let E divide AC in the ratio 1 : 2 and F divide AC in the ratio 2 : 1
therefore position vector of E is fraction numerator 1 left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis space plus space 2 left parenthesis stack 0 right parenthesis with rightwards arrow on top over denominator 1 plus 2 end fraction space equals space fraction numerator straight b with rightwards arrow on top space plus space straight d with rightwards arrow on top over denominator 3 end fraction
Position vector of point dividing PD in the ratio 1:2 is fraction numerator left parenthesis 1 right parenthesis space straight d with rightwards arrow on top space plus left parenthesis 2 right parenthesis open parentheses begin display style fraction numerator straight b with rightwards arrow on top over denominator 2 end fraction end style close parentheses over denominator 1 plus 2 end fraction space equals fraction numerator straight b with rightwards arrow on top plus straight d with rightwards arrow on top over denominator 3 end fraction
Position vector of straight F with rightwards arrow on top is fraction numerator 2 left parenthesis straight b with rightwards arrow on top plus straight d with rightwards arrow on top right parenthesis space plus space 1 left parenthesis 0 with rightwards arrow on top right parenthesis over denominator 1 plus 2 end fraction space equals space fraction numerator 2 straight b with rightwards arrow on top plus 2 straight d with rightwards arrow on top over denominator 3 end fraction
Position vector of point dividing BQ in the ratio 2 : 1 is fraction numerator 2 space open parentheses straight d with rightwards arrow on top plus begin display style fraction numerator straight b with rightwards arrow on top over denominator 2 end fraction end style close parentheses plus space 1 left parenthesis straight b with rightwards arrow on top right parenthesis over denominator 2 plus 1 end fraction equals space fraction numerator 2 straight b with rightwards arrow on top space plus space 2 straight d with rightwards arrow on top over denominator 3 end fraction

 ∴    DP and BQ cut diagonal AC in its points of trisection which are also the points of trisection of DP and BQ respectively.
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127. Points E and F are taken on the sides BC and CD of a parallelogram ABCD such that open vertical bar BF with rightwards arrow on top close vertical bar thin space colon thin space open vertical bar FC with rightwards arrow on top close vertical bar space equals space straight mu comma space space space space open vertical bar DE with rightwards arrow on top close vertical bar space colon thin space open vertical bar EC with rightwards arrow on top close vertical bar space equals space straight lambdaThe straight lines FD and AE intersect at the point O. Find the ratio open vertical bar FO with rightwards arrow on top close vertical bar space colon thin space open vertical bar OD with rightwards arrow on top close vertical bar.
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128. Prove that the vertices straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space minus space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top and straight c with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space 6 space straight k with hat on top can form the sides of a triangle. Find the lengths of the medians of the triangle.
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 Multiple Choice QuestionsShort Answer Type

129. Evaluate the scalar product:
left parenthesis 3 straight a with rightwards arrow on top space minus space 5 space straight b with rightwards arrow on top right parenthesis. space space left parenthesis 2 straight a with rightwards arrow on top plus space 7 straight b with rightwards arrow on top right parenthesis
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130.

Find open parentheses straight b with rightwards arrow on top space minus space straight a with rightwards arrow on top close parentheses. space space open parentheses 3 space straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top close parentheses when
    straight a with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space straight b with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space straight j with hat on top space minus space space 3 space straight k with hat on top

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