triangle ABC and triangle DBC are on the same base and on opposite sides of BC and O is the point of intersection of AD and BC.prove that 
Area(ABC)/Area(DBC)=AO/DO.

Asked by pugalatha | 12th Sep, 2015, 10:21: 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.
L e t space M space a n d space N space a r e space t h e space f o o t space o f space p e r p e n d i c u l a r s space f r o m space t h e space v e r t i c e s space A space a n d space D space o n space t h e space b a s e space B. S i n c e space A M space a n d space D N space a r e space p e r p e n d i c u l a r s space o n space t h e space s a m e space b a s e space B C comma space A M space i s space p a r a l l e l space t o space D N. L e t space theta equals space angle M A O equals angle N D O rightwards double arrow A M equals A O c o s theta space a n d space D N equals D O cos theta W e space k n o w space t h a t space t h e space a r e a space o f space t h e space t r i a n g l e space A B C equals 1 half cross times b a s e cross times a l t i t u d e T h u s comma space a r e a space o f space t h e space t r i a n g l e space A B C equals 1 half B C cross times A M equals 1 half B C cross times A O c o s theta space space... left parenthesis 1 right parenthesis S i m i l a r l y comma space t h e space a r e a space o f space t h e space t r i a n g l e space B C D equals 1 half B C cross times D N equals 1 half B C cross times D O c o s theta space space... left parenthesis 2 right parenthesis D i v i d i n g space e q u a t i o n space left parenthesis 1 right parenthesis space b y space left parenthesis 2 right parenthesis comma space w e space h a v e comma fraction numerator A r e a open parentheses triangle A B C close parentheses over denominator A r e a open parentheses triangle D B C close parentheses end fraction equals fraction numerator 1 half B C cross times A O c o s theta over denominator 1 half B C cross times D O c o s theta end fraction rightwards double arrow fraction numerator A r e a open parentheses triangle A B C close parentheses over denominator A r e a open parentheses triangle D B C close parentheses end fraction equals fraction numerator A O over denominator D O end fraction H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 14th Sep, 2015, 12:31: PM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.