The diagonals of a quadrilateral ABCD  are perpendiculars . Show that quadrilateral formed by joining the midpoints of its sides is a rectangle

Asked by Paresh | 11th Dec, 2015, 03:00: 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.
 
A B C D space i s space a space q u a d r i l a t e r a l. P comma space Q comma space R space a n d space S space a r e space t h e space m i d p o i n t s space o f space t h e space s i d e s space A B comma space B C comma space C D space a n d space D A space r e s p e c t i v e l y. G i v e n space t h a t space t h e space d i a g o n a l space A C space a n d space B D space i n t e r s e c t space a t space 90 degree W e space n e e d space t o space p r o v e space t h a t space P Q R S space i s space a space r e c tan g l e.  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 P space i s space t h e space m i d p o i n t space o f space A B space a n d space S space i s space t h e space m i d p o i n t space o f space D A  M i d p o i n t space t h e o r e m colon T h e space l i n e space s e g m e n t space c o n n e c t i n g space t h e space m i d p o i n t s space o f space t w o space s i d e s space o f space a space t r i a n g l e space i s space p a r a l l e l space t o space t h e space t h i r d space s i d e space a n d space i s space e q u a l space t o space o n e space h a l f space o f space t h e space t h i r d space s i d e.  T h u s comma space P S space i s space p a r a l l e l space t o space B D space a n d space P S equals 1 half B D... left parenthesis 1 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 B C D colon R space i s space t h e space m i d p o i n t space o f space C D space a n d space Q space i s space t h e space m i d p o i n t space o f space B C  B y space m i d p o i n t space t h e o r e m comma space space R Q space i s space p a r a l l e l space t o space B D space a n d space R Q equals 1 half B D... 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 C D colon S space i s space t h e space m i d p o i n t space o f space A D space a n d space R space i s space t h e space m i d p o i n t space o f space C D  B y space m i d p o i n t space t h e o r e m comma space space S R space i s space p a r a l l e l space t o space A C space a n d space S R equals 1 half A C... left parenthesis 3 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 C colon P space i s space t h e space m i d p o i n t space o f space A B space a n d space Q space i s space t h e space m i d p o i n t space o f space B C  B y space m i d p o i n t space t h e o r e m comma space space P Q space i s space p a r a l l e l space t o space A C space a n d space P Q equals 1 half A C... left parenthesis 4 right parenthesis F r o m space e q u a t i o n s space left parenthesis 1 right parenthesis comma space left parenthesis 2 right parenthesis comma space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space h a v e comma space P Q equals R S equals 1 half A C space a n d space P S equals R Q equals 1 half B D T h u s space w e space h a v e space p r o v e d space t h e space o p p o s i t e space s i d e s space o f space t h e space q u a d r i l a t e r a l space P Q R S space a r e space e q u a l. N o w space 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 c o r n e r space a n g l e s space o f space t h e space q u a d r i l a t e r a l space P Q R S equals 90 degree  G i v e n space t h a t space A C space a n d space B D space i n t e r s e c t space a t space 90 degree A l s o space w e space h a v e space p r o v e d space t h a t space P S parallel to B D space a n d space R Q parallel to B D S i m i l a r l y comma space w e space h a v e comma space P Q parallel to A C space a n d space S R parallel to A C T h u s comma space t h e space c o r n e r space a n g l e s space o f space t h e space q u a d r i l a t e r a l a l space P Q R S space a r e space 90 degree. H e n c e space P Q R S space i s space a space r e c tan g l e.

Answered by Vimala Ramamurthy | 12th Dec, 2015, 01:26: PM