Question

If the diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral. Then the quadrilateral is a
a) a kite b) a rhombus c) a square d) N-O-T
​

Answer 1

Answer:

Given−

ABCDisacyclicquadrilateral.

ThediagonalsAC&BDarethediametersofacircle

whichpassesthroughthepointsA,B,C&D.

TofindoutthekindofthequadrilateralABCD.

Solution−

ACisthediameterofthecirclewhichpassesthrough

thepointsA,B,C&D.

∠ABCisanangleinthesemicircle.

i.e∠ABC=90

o

.

ButABCDisacyclicquadrilateral.

∴∠ABC+∠ADC=180

o

⟹∠ADC=180

o

−∠ABC=180

o

−90

o

=90

o

.

Similarly

BDisthediameterofthecirclewhichpassesthrough

thepointsA,B,C&D.

∠DABisanangleinthesemicircle.

i.e∠DAB=90

o

.

ButABCDisacyclicquadrilateral.

∴∠DAB+∠DCB=180

o

⟹∠DCB=180

o

−∠DAB=180

o

−90

o

=90

o

.

So∠ABC=∠DCB=∠ADC=∠DAB=90

o

.

Thisconfirmswiththedefinitionofarectangle.

∴ABCDisarectangle.