InterviewSolution
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InterviewSolution
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The diagonals of a quadrilateral ABCD are mutually perpendicular . Prove that the quadrilateral formed by joining the mid-points of its consecutive sides is a reactangle. |
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Answer» Let P, Q , R and S are the mid-points of the sides AB, BC,CD and DA respectively of `square ABCD`. In `triangleBAC` , Since P is mid point of AB and Q is mid-point of BC `therefore PQ"||"AC and PQ=(1)/(2)AC` (mid-point theorem)…(1) Similarly `RS"||"AC and RS =(1)/(2)AC` (mid point theorem)...(2) Form (1) and (2) , we get `PQ"||"RS and PQ=RS` `implies square PQRS` is a parallelogram. (`because` one pair of opposite sides is equal and parallel.) Let the diagonals AC and BD of `square ABCD` intereset each other at point O. In `triangleABD`, P is the mid point of AB and S is the mid-point of AD. `therefore PS"||"BD` `implies PN"||"MO` Similarly, `PQ"||"AC` `implies PM"||"NO` `therefore sqaure PMON` is a parallelogram `" "` (pairs of opposite sides are parallel) `implies angle MPN =angle MON` `implies angle MPN=angleBOA` `implies angleMPN=90^(@)" "(because angleBOA=90^(@), AC bot BD)` `implies angleQPS=90^(@)` Now, `squarePQRS` is a parallelogram and its one `angle is `90^(@)` `therefore sqaure PQRS` is a rectangle. Hence proved. |
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