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# ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the mid-points of AB, AC, CD and BO respectively, show that PQRS is a rhombus.

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Solution

## Given, ABCD is a quadrilateral in which AD = BC and P, Q, R, S are the mid points of AB, AC, CD, BD, respectively. To prove, PQRS is a rhombus In ΔABC, P and Q are the mid points of the sides AB and AC respectively By the midpoint theorem, we get, PQ∥BC, PQ = 1/2 BC. ---(i) In ΔADC, Q and R are the mid points of the sides AC and DC respectively By the mid point theorem, we get, QR∥AD and QR = 1/2 AD = 1/2 BC (AD = BC) ---(ii) In ΔBAD, By the mid point theorem, we get, PS∥AD and PS = 1/2 AD (AD = BC) ---(iii) In ΔBCD, R and S are the mid points of the sides CD and BD respectively By the midpoint theorem, we get, RS∥BC and SR = 1/2 BC (AD = BC) ---(iv) From above eqns. PQ = QR = RS = PS Thus, PQRS is a rhombus.  Suggest Corrections  2      Similar questions  Related Videos   The Mid-Point Theorem
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