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7. If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
Answer :

Given: Diagonals NP and QM of a cyclic quadrilateral are diameters of the circle through the vertices M, P, Q and N of the quadrilateral NQPM.
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To prove: Quadrilateral NQPM is a rectangle.

Proof: ON = OP = OQ = OM ...(Radii of circle)
Now, ON = OP = (1/2) NP
and OM = OQ = (1/2) MQ
Therefore, NP = MQ
Hence, the diagonals MPQN are equal and bisect each other.
Hence, it is proved that quadrilateral NQPM is a rectangle.