Question
Fri March 20, 2015 By: Rrdave2611 Dave

state mid point theoram
 
 
 
 
 

Expert Reply
Vimala Ramamurthy
Sun March 22, 2015
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.
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 j o i n i n g space t h e space m i d p o i n t s space o f space t h e 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 h a l f space t h e space l e n g t h space o f space t h e space t h i r d space s i d e. P r o o f colon D r a w space a space l i n e space D F space s u c h space t h a t space D E equals E F space a n d space j o i n space F C 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 s space triangle E A D space a n d space triangle E C F  A E equals E C space space space space space space space space space space space space space space space left square bracket E space i s space t h e space m i d p o i n t space o f space t h e space s i d e space A C right square bracket angle A E D equals angle C E F space space space space left square bracket v e r t i c a l l y space o p p o s i t e space a n g l e s space a r e space e q u a l right square bracket D E equals E F space space space space space space space space space space space space space space space space left square bracket b y space c o n s t r u c t i o n right square bracket 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 therefore triangle A 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 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. therefore angle A D E equals angle C F E B u t space t h e s e space a r e space a s l t e r n a t e space i n t e r i o r space a n g l e s space a n d space h e n c e comma space B D parallel to F C comma rightwards double arrow B C F D space i s space a space p a r a l l e log r a m rightwards double arrow D E parallel to B C  N o w space c o n s i d e r space t h e space s i d e space D E colon D F equals B C rightwards double arrow D F equals D E plus E F rightwards double arrow D F equals 2 D E rightwards double arrow B C equals 2 D E rightwards double arrow D E equals 1 half B C
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