abcd is a square and the diagonals intersect at o.if p is a point on AB such that AO=AP prove that 3POB=AOP 

Asked by rgkgmu | 21st Dec, 2015, 07:41: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
I n space a space s q u a r e space d i a g o n a l s space b i s e c t space t h e space a n g l e s. T h u s comma space angle O A B equals angle O B A equals fraction numerator 90 degree over denominator 2 end fraction equals 45 degree A l s o space g i v e n space t h a t space O A equals A P space a n d space h e n c e space triangle A O P space i s space a n space i s o s c e l e s space t r i a n g l e. rightwards double arrow space angle A O P equals angle A P O B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle A O P plus angle A P O plus angle P A O equals 180 degree rightwards double arrow angle A O P plus angle A O P plus angle B A O equals 180 degree rightwards double arrow 2 angle A O P plus 45 degree equals 180 degree rightwards double arrow 2 angle A O P equals 180 degree minus 45 degree rightwards double arrow angle A O P equals fraction numerator 180 degree minus 45 degree over denominator 2 end fraction T h u s comma space angle A O P equals angle A P O equals fraction numerator 180 degree minus 45 degree over denominator 2 end fraction equals 67.5 degree angle A O P space a n d space angle B O P space a r e space c o m p l e m e n t a r y space a n g l e s rightwards double arrow angle A O P plus angle B O P equals 90 degree rightwards double arrow 67.5 degree plus angle B O P equals 90 degree rightwards double arrow angle B O P equals 90 degree minus 67.5 degree equals 22.5 degree rightwards double arrow 3 angle B O P equals 3 cross times 22.5 degree rightwards double arrow 3 angle B O P equals 67.5 degree  rightwards double arrow 3 angle B O P equals angle A O P

Answered by Vimala Ramamurthy | 22nd Dec, 2015, 04:32: PM