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CBSE Class 9 Answered

In a triangle ABC, with side AB = AC. D, E and F be the midpoints of sides BC, CA and AB respectively. Show that the line segment AD is perpendicular to the line segment EF and is bisected by it

Asked by Paresh | 11 Dec, 2015, 02:52: PM
answered-by-expert 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 i d e r space t h e space t r i a n g l e space A B C G i v e n space t h a t space A B equals A C comma space h e n c e space triangle A B C space i s space a n space i s o s c e l e s space t r i a n g l e. T h u s comma space t h e space b a s e space a n g l e s space angle A B C space a n d space angle A C B space a r e space e q u a l. A l s o space g i v e n space t h a t space 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 C space a n d space A B space r e s p e c t i v e l y. 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 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 c o n g r u e n t 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. space T h u s comma space E F space i s space p a r a l l e l space t o space B C space a n d space E F equals 1 half B C 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 F E space i s space p a r a l l e l space t o space B C space a n d space F space i s space m i d p o i n t space o f space A B rightwards double arrow F M space i s space p a r a l l e l space t o space B C rightwards double arrow F M space i s space p a r a l l e l space t o space B D  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 T h e space s t r a i g h t space l i n e 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 o n e 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 a n o t h e r comma space b i s e c t s space t h e space t h i r d space s i d 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 A B space a n d space F M space i s space p a r a l l e l space t o space B D 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 A D.  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 F E space i s space p a r a l l e l space t o space B C space a n d space E space i s space m i d p o i n t space o f space A C rightwards double arrow E M space i s space p a r a l l e l space t o space B C rightwards double arrow E M space i s space p a r a l l e l space t o space D C rightwards double arrow E space i s space t h e space m i d p o i n t space o f space A C space a n d space E M space i s space p a r a l l e l space t o space D C 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 A D.  T h u s comma space F M equals E M rightwards double arrow A D space i s space t h e space b i s e c t o r space o f space E F  T h e space l i n e space j o i n i n g space t h e space v e r t e x space t o space i t s space o p p o s i t e space s i d e space i n space a n space i s o s c e l e s space t r i a n g l e i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space t h i r d space s i d e. S i n c e space triangle A B C space i s space a n space i s o s c e l e s space t r i a n g l e space a n d space A D space i s space t h e space l i n e space f r o m space A comma space A D space i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space B C. M space l i e s space o n space t h e space l i n e space A D rightwards double arrow A D space i s space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space E F.
Answered by Vimala Ramamurthy | 12 Dec, 2015, 08:53: AM
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