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ABCD is a rectangle in which diagonal BD bisects <B. Show that ABCD is a square

Asked by Paresh | 11 Dec, 2015, 02:32: PM
answered-by-expert 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.
W e space k n o w space t h a t space o p p o s i t e space s i d e s space o f space a space r e c tan g l e space a r e space e q u a l. T h u s comma A B equals C D space a n d A D equals B C F o r space a space r e c tan g l e space a l l space t h e space a n g l e s space a r e space e q u a l space t o space 90 degree rightwards double arrow angle B equals 90 degree S i n c e space B D space i s space t h e space a n g l e space b i s e c t o r comma space angle A B D equals angle D B C equals fraction numerator 90 degree over denominator 2 end fraction equals 45 degree... left parenthesis 1 right parenthesis S i m i l a r l y space D B space i s space t h e space a n g l e space b i s e c t o r space a n d space h e n c e comma space angle A D B equals angle C D B equals fraction numerator 90 degree over denominator 2 end fraction equals 45 degree... left parenthesis 2 right parenthesis N o w space c o n s i d e r space t h e space t r i a n g l e space triangle A B D colon F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma angle A B D equals angle A D B equals 45 degree T h e space b a s e space a n g l e s space o f space triangle A B D space a r e space e q u a l. H e n c e space triangle A B D 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 space a r e space e q u a l comma space w i t h space v e r t e x space a t space A. T h u s comma space A B equals A D B u t space w e space h a v e comma space A B equals C D space a n d space A D equals B C S o comma space w e space h a v e comma space A B equals C D equals A D equals B C. T h e r e f o r e space A B C D i s space a space s q u a r e.
Answered by Vimala Ramamurthy | 12 Dec, 2015, 05:44: AM

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