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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 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

10

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.


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