4. The vertices of a parallelogram lie on a circle. Prove that its diagonals are equal.

Asked by avantikadandapani | 17th Nov, 2014, 06:47: PM

Expert Answer:

G i v e n space t h a t space t h e space v e r t i c e s space o f space a space p a r a l l e log r a m space l i e space o n space a space c i r c l e. T h a t space i s space i t space i s space a space c y c l i c space p a r a l l e log r a m space a n d space h e n c e space t h e space o p p o s i t e space a n g l e s space a r e space s u p p l e m e n t a r y. I n space a space p a r a l l e log r a m comma space o p p o s i t e space a n g l e s space a r e space e q u a l space a n d space a d j a c e n t space a n g l e s space a r e space s u p p l e m e n t a r y. L e t space x space a n d space y space b e space t h e space a d j a c e n t space a n g l e s space o f space t h e space p a r a l l e log r a m S i n c e space i t space i s space c y c l i c comma space w e space h a v e comma x plus x equals 180 degree rightwards double arrow 2 x equals 180 degree rightwards double arrow x equals fraction numerator 180 degree over denominator 2 end fraction equals 90 degree I n space a space p a r a l l e log r a m comma space i f space o n e space o f space t h e space a n g l e space i s space 90 degree comma space t h e n space i t space i s space a space r e c tan g l e. I n space a space r e c tan g l e comma space d i a g o n a l s space a r e space e q u a l space a n d space h e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 18th Nov, 2014, 08:45: AM