Points A and B are on the same sides of a line l. AD and BE are perpendiculars to l, meeting l at D and E respectively. C is the midpoint of AB. Prove that CD = DE

Asked by Paresh | 11th Dec, 2015, 02:45: 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.
 
C o n s t r u c t space a space l i n e space C F space p a r a l l e l space t o space A D space a n d space h e n c e space p a r a l l e l space t o space B E. T h e r e f o r e comma space A D comma space C F space a n d space B E space a r e space t h r e e space p a r a l l e l space l i n e s.  C o n s i d e r space t h e space f o l l o w i n g space t h e o r e m colon I f space t h r e e space o r space m o r e space p a r a l l e l space l i n e s space i n t e r s e c t space t w o space t r a n s v e r s a l s comma space t h e n space t h e y space c u t space o f f space t h e space t r a n s v e r s a l s space p r o p o r t i o n a l l y.  T h u s comma fraction numerator C A over denominator C B end fraction equals fraction numerator F D over denominator F E end fraction equals 1 space space space open square brackets because C space i s space t h e space m i d p o i n t close square brackets rightwards double arrow F D equals F E rightwards double arrow F space i s space t h e space m i d p o i n t space o f space D E C o n s i d e r space t h e space t r i a n g l e s space triangle C D E space a n d space triangle C E F F D equals F E space space open square brackets because F space i s space t h e space m i d p o i n t space o f space D E close square brackets angle C F D equals angle C F E equals 90 degree space space space open square brackets because A D comma space C F space a n d space B E space a r e space t h r e e space p a r a l l e l space l i n e s space a n d space A D comma space C F comma space B E space perpendicular space t o space l i n e space l close square brackets C F equals C F space space space space open square brackets c o m m o n close square brackets 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 space triangle C D E approximately equal to triangle C E F 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 r r e s p o n d i n g space t r i a n g l e s space a r e space c o n g r u e n t. therefore C D equals C E H e n c e space p r o v e d. P l e a s e space c h e c k space t h e space q u e s t i o n colon Q u e s t i o n space s h o u l d space b e colon P o i n t s space A space a n d space B space a r e space o n space t h e space s m a e space s i d e s space o f space a space l i n e. space A D space a n d space B E space a r e space p e r p e n d i c u l a r s space t o space l comma space m e e t i n g l space a t space D E space a n d space E space r e s p e c t i v e l y. space C space i s space t h e space m d i p o i n t space o f space A B. space P r o v e space t h a t space C D equals C E.

Answered by Vimala Ramamurthy | 12th Dec, 2015, 08:08: AM