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 , 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 is parallel to 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.
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127.Points E and F are taken on the sides BC and CD of a parallelogram ABCD such that . The straight lines FD and AE intersect at the point O. Find the ratio
Long Answer Type
, where O is any arbitrary point.
is parallel to 
where K divides AB in the ratio 1 : 3.
. The straight lines FD and AE intersect at the point O. Find the ratio 
Short Answer Type
and 
can form the sides of 
respectively.
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