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Question
ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus.
Solution
In Δ ABC, P and Q are mid points of AB and AC respectively.
So, PQ || BC, and PQ = `1/2 `BC ..................(1)
Similarly, in ΔADC, .................(2)
Now, in ΔBCD, SR = `1/2` BC .........................(3)
Similarly, in ΔABD, PS = `1/2` AD=`1/2` BC ...............(4)
Using (1), (2), (3), and (4).
PQ = QR = SR = PS
Since, all sides are equal
Hence, PQRS is a rhombus.
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