if the diagonals of a quadilateral are equal and bisect at right angles then prove it is a square

Asked by rgkgmu | 21st Dec, 2015, 07:37: PM

Expert Answer:

G i v e n space t h a t space t h e space d i a g o n a l s space o f space t h e space q u a d r i l a t e r a l space a r e space e q u a l space a n d space t h e y space b i s e c t space a t space r i g h t space a n g l e s. W e space n e e d space t o space p r o v e space t h a t space t h e space q u a d r i l a t e r a l space i s space a space s q u a r e. L e t space u s space o b s e r v e space t h e space f o l l o w i n g space f i g u r e.
G i v e n space t h a t space A C equals B D A n d space O A equals O C space a n d space O B equals O D A l s o space w e space h a v e comma space A C perpendicular B D  C o n s i d e r space t h e space t r i a n g l e s comma space triangle A D O space a n d space triangle C B O O A equals O C angle A O D equals angle C O B equals 90 degree O D equals O B T h u s comma space b y space S i d e minus A n g l e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space w e space h a v e comma triangle A D O space a n d space triangle C B O space a r e space c o n g r u e n t. T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r u e n t. T h u s space A D equals B C space space a n d space angle D A O equals angle B C O  S i m i l a r l y comma space C o n s i d e r space t h e space t r i a n g l e s comma space triangle A O B space a n d space triangle C O D O A equals O C angle A O B equals angle C O D equals 90 degree O D equals O B T h u s comma space b y space S i d e minus A n g l e minus S i d e space c r i t e r i o n space o f space c o n g r u e n c e comma space w e space h a v e comma triangle A O B space a n d space triangle C O D space a r e space c o n g r u e n t. T h e space c o r r e s p o n d i n g space p a r t s space o f space t h e space c o n g r u e n t space t r i a n g l e s space a r e space c o n g r u e n t. T h u s space A B equals C D space space a n d space angle O A B equals angle O C D  A B equals C D space a n d space A D equals B C A C space a n d space B D space b i s e c t i n g space a t space r i g h t space a n g l e s. space rightwards double arrow A B C D space i s space a space r h o m b u s space  A B C D space i s space a space r h o m b u s space a n d space A C equals B D rightwards double arrow A B C D space i s space a space s q u a r e.

Answered by Vimala Ramamurthy | 22nd Dec, 2015, 10:26: AM

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