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.ABCD is a quadalateral in wch RQ R and S armid poiats of the sides AB, BO, ED and DA(sep Fig 8 29) AC is a diagenai Show that(O SRAC and SR ACFig 8 2 |
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Answer» Given: ABCD is a quadrilateral in which P, Q, R and S are mid-points of the sides AB, BC, CD and DA. AC is a diagonal. To Prove: (1 SR || AC and SR = AC (2) PQ = SR (3) PQRS is a parallelogram. Proof:(1) In Δ DAC,∴ S is the mid-point of DA and R is the mid-point of DC∴ SR || AC and SR = ½AC [Mid-point theorem] (2) In Δ BAC,∴ P is the mid-point of AB and Q is the mid-point of BC∴ PQ || AC and PQ = ½AC [Mid-point theorem]But from (1) SR = AC∴ PQ = SR (3)PQ || AC ..............From (2)SR || AC ..................From(1)∴ PQ || SR [Two lines parallel to the same line are parallel to each other]Also, PQ = SR ................From (2)∴ PQRS is a parallelogram. |
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