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Question

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


Solution


Let ABCD be a cyclic quadrilateral having diagonals BD and AC, intersecting each other at point O which is centre of the circle.

Consider, ΔABC,as AC is the diameter, therefore,

ABC=90 (Angle in a semi circle is 90)

Consider, ΔBCD,as DB is the diameter, therefore,

DCB=90 (Angle in a semi circle is 90)

Since, ABCD be a cyclic quadrilateral, therefore,

ADC=180ABC=90 and 

DAB=180DBC=90 

Since, all the angles of cyclic quadrilateral ABCD are 90, therefore, 

ABCD is a rectangle.


Mathematics
DUMMY
Standard IX

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