ABCD is a rhombus, and M and N are the mid points of AD and AB respectively. The diagonal AC and BD intersect at O. If ar(AMN) = 100 cm², then find the area of the polygon MNBCD

Asked by tv2000in | 28th Jul, 2017, 04:58: PM

Expert Answer:

begin mathsize 16px style
L e t space A C space a n d space M N space m e e t space a t space p o i n t space Q. space
A C perpendicular D B space left parenthesis D i a g o n a l s space o f space a space r h o m b u s space b i s e c t space e a c h space o t h e r right parenthesis...... left parenthesis 1 right parenthesis

I n space increment A Q N space a n d space increment A Q M comma
A M equals A N space space left parenthesis M space a n d space N space a r e space m i d p o i n t s semicolon space s i d e s space o f space a space r h o m b u s space a r e space e q u a l right parenthesis
A Q equals A Q space left parenthesis c o m m o n space s i d e right parenthesis
angle M A Q equals angle N A Q space
therefore increment A Q N space approximately equal to space increment A Q M space left parenthesis S A S space t e s t right parenthesis
angle A Q N equals angle A Q M space left parenthesis C A C T right parenthesis
angle A Q N plus angle A Q M equals 180 degree space left parenthesis a n g l e s space f o r m i n g space a space l i n e a r space p a i r right parenthesis
2 angle A Q N equals 180 degree
therefore angle A Q N equals 90 degree space space space........ left parenthesis 2 right parenthesis

F r o m space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis
M N parallel to D B...... left parenthesis 3 right parenthesis


I n space increment A M N space a n d space increment A D B comma
angle M A N equals angle D A B.... c o m m o n space a n g l e
angle A M N equals angle A D B space..... f r o m space left parenthesis 3 right parenthesis space a n d space c o r r e s p o n d i n g space a n g l e s
therefore increment A M N tilde space increment A D B space b y space A A space t e s t

therefore fraction numerator A left parenthesis triangle A M N right parenthesis over denominator A left parenthesis triangle A D B right parenthesis end fraction equals fraction numerator A N squared over denominator A B squared end fraction space space space space....... T h e o r e m space o n space a r e a s space o f space s i m i l a r space t r i a n g l e s
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A N squared over denominator left parenthesis 2 A N right parenthesis squared end fraction space space space space space space space left parenthesis N space i s space t h e space m i d p o i n t space o f space A B right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator A N squared over denominator 4 A N squared end fraction
space space space space space fraction numerator 100 over denominator A left parenthesis triangle A D B right parenthesis end fraction equals 1 fourth

A left parenthesis triangle A D B right parenthesis equals 400 c m squared

therefore A left parenthesis r h o m b u s space A B C D right parenthesis equals space 2 cross times 400 equals 800 c m squared

A left parenthesis ▭ M N B C D right parenthesis equals A left parenthesis r h o m b u s space A B C D right parenthesis minus A left parenthesis triangle A M N right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 800 minus 100 equals 700 c m squared
A left parenthesis ▭ M N B C D right parenthesis equals 700 c m squared



end style

Answered by Rashmi Khot | 31st Jul, 2017, 03:21: PM