Question
Sat March 14, 2015 By: Archita Srivastava

ABCD is a trapezium with parallel sides AB=a cm and DC=b cm. E and F are the midpts of the non parallel sides. The ratio of ar(ABFE) and ar(EFCD) is

Expert Reply
Prasenjit Paul
Sun March 15, 2015
S i n c e space E space a n d space t h e space m i d p o i n t s space o f space A D space a n d space B C comma space t h e r e f o r e space E F space s h o u l d space b e space p a r a l l e l space t o space D C space a n d space A C. I n space increment A B D comma E 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 i s space p a r a l l e l space t o space A B comma space t h e r e f o r e space i t space s h o u l d space i n t r e s e c t space B D thin space a t space t h e space m i d p o i n t space G space a n d space E G equals 1 half A B therefore E G equals a over 2 I n space increment B D C comma F space i s space t h e space m i d p o i n t space o f space B C space a n d space i s space p a r a l l e l space t o space D C comma space t h e r e f o r e space i t space s h o u l d space i n t r e s e c t space B D thin space a t space t h e space m i d p o i n t space G space a n d space G F equals 1 half D C therefore D C equals b over 2 S i n c e space E space a n d space F 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 B C comma space t h e r e f o r e space d i s tan c e space b e t w e e n space A B space & space E F comma space a n d space E F space & space D C space s h o u l d space b e space s a m e space a n d w h i c h space i s space e q u a l space t o space h. space A r e a space o f space Q u a d. A B F E equals A r. space Q u a d. space A B G E plus A r. space increment space B F G equals 1 half open parentheses a over 2 plus a close parentheses cross times h plus 1 half cross times b over 2 cross times h equals 1 half cross times h cross times open parentheses fraction numerator 3 a over denominator 2 end fraction plus b over 2 close parentheses A r e a space o f space Q u a d. D E F C equals A r. space Q u a d. space G F C D plus A r. space increment space D E G equals 1 half open parentheses b over 2 plus b close parentheses cross times h plus 1 half cross times b over 2 cross times h equals 1 half cross times h cross times open parentheses a over 2 plus fraction numerator 3 b over denominator 2 end fraction close parentheses fraction numerator A r. Q u a d. A B F E space over denominator A r. Q u a d. D E F C end fraction equals fraction numerator 3 a plus b over denominator a plus 3 b end fraction
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