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Question

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

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Solution

## Given: ABCD is 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. Proof: In ∆ABC, P and Q are the midpoints of the sides AB and AC respectively. By the Mid point theorem, we get PQ || BC and PQ = $\frac{1}{2}$BC ...(1) In ∆ADC, Q and R are the midpoints of the sides AC and DC respectively. By the Mid point theorem, we get QR || AD and QR = $\frac{1}{2}$AD = $\frac{1}{2}$BC (Since AD = BC) ...(2) Similarly, in ∆BCD, we have RS || BC and RS = $\frac{1}{2}$BC ...(3) In ∆BAD, we have PS || AD and PS = $\frac{1}{2}$AD = $\frac{1}{2}$BC (Since AD = BC) ...(4) From the equations (1), (2), (3), (4), we get PQ = QR = RS = RS Thus, PQRS is a rhombus.

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