Question

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

Answer 1

Given that,

A cyclic quadrilateral .

$ABCD$,AC and $BD$$ are diameters of the circle where they meet at center O of the circle.

To prove:$ABCD$ is a rectangle.

Proof: In triangle $\Delta AOD$ and $\Delta BOC$,

$OA=OC$ (both are radii of same circle)

$\angle AOD=\angle BOC$ (vert.opp$\angle S$)

$OD=OB$(both are radii of same circle)

$\therefore$ $\Delta AOD\cong \Delta BOC\Rightarrow AD=BC(C.P.C.T)$

Similarly,by taking $\Delta AOB$ and $\Delta COD$,$AB=DC$

Also, $\angle BAD=\angle ABC=\angle BCD=\angle ADC={90}^0$(angle in a semicircle)

$\therefore$ $ABCD$ is a rectangle.