In a parallelogram ABCD, E and F are the midpoints of sides AB and CD. Prove that the line segments AF and CE trisect the diagonal BD

Asked by Paresh | 11th Dec, 2015, 01:38: 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.
E space a n d space F space a r e space t h e space m i d p o i n t s space o f space A B space a n d space C D. 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 l i n e space s e g m e n t s space A F space a n d space C E space t r i s e c t space t h e space d i a g o n a l space B D. T h a t space i s space w e space n e e d space t o space p r o v e space t h a t space B M equals M N equals N D. C o n s i d e r space t h e space s i d e s space A B space a n d space C D colon A B equals A E plus E B space a n d space C D equals D F plus C F S i n c e space A B equals C D comma space w e space h a v e comma A E plus E B equals D F plus C F T h e space l i n e space E F space i s space p a r a l l e l space t o space A D space a n d space B C. T h e r e f o r e comma space A E equals E B equals fraction numerator A B over denominator 2 end fraction equals space D F equals C F equals fraction numerator C D over denominator 2 end fraction rightwards double arrow A E C F space i s space a space p a r a l l e l o g r a m.  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 N A E C F space i s space a space p a r a l l e log r a m. rightwards double arrow c o r r e s p o n d i n g space a n g l e s space a r e space e q u a l. rightwards double arrow angle F A E equals angle C E B rightwards double arrow A N space a n d space C M space a r e space p a r a l l e l C o n v e r s e space o f space m i d p o i n t space T h e o r e m colon I f space a space l i n e space i s space d r a w n space t h r o u g h space t h e space m i d minus p o i n t space o f space a space s i d e space o f space a space t r i a n g l e space p a r a l l e l space t o space t h e space s e c o n d space s i d e comma space i t space w i l l space b i s e c t space t h e space t h i r d space s i d e. B y space c o n v e r s e space o f space m i d p o i n t space t h e o r e m comma space M space i s space t h e space m i d p o i n t space o f space B N. B N equals B M plus M N rightwards double arrow B N equals 2 M N rightwards double arrow M N equals B M equals fraction numerator B N over denominator 2 end fraction... left parenthesis 1 right parenthesis  C o n s i d e r space t h e space t r i a n g l e space triangle C D M A E C F space i s space a space p a r a l l e l o g r a m. rightwards double arrow c o r r e s p o n d i n g space a n g l e s space a r e space e q u a l. rightwards double arrow angle M C F equals angle A F D rightwards double arrow C M space a n d space F N space a r e space p a r a l l e l C o n v e r s e space o f space m i d p o i n t space T h e o r e m colon I f space a space l i n e space i s space d r a w n space t h r o u g h space t h e space m i d minus p o i n t space o f space a space s i d e space o f space a space t r i a n g l e space p a r a l l e l space t o space t h e space s e c o n d space s i d e comma space i t space w i l l space b i s e c t space t h e space t h i r d space s i d e. B y space c o n v e r s e space o f space m i d p o i n t space t h e o r e m comma space N space i s space t h e space m i d p o i n t space o f space D M. D M equals D N plus M N rightwards double arrow D M equals 2 M N rightwards double arrow M N equals D N equals fraction numerator D M over denominator 2 end fraction... left parenthesis 2 right parenthesis 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 B M equals D N equals M N

Answered by Vimala Ramamurthy | 11th Dec, 2015, 06:58: PM