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In a rectangle ABCD of length 14 and breadth 7, an arc with centre A and radius 14 is drawn. Find the area of shaded region.
 

 Jan 8, 2020
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Let  E   be the  point on the  circle that also lies on DC

 

Then     DE  =   sqrt  ( 14^2 - 7^2)  = sqrt [ 7^2  ( 2^2  - 1) ] =  7sqrt (3)

 

So  the  area of triangle  DEA =  (1/2) (DE)(DA)  = (1/2) (7sqrt (3)) ( 7)   = (49/2)sqrt (3)  units^2    (1)

 

And the sine of  angle  AED  = sine of  angle  EAB  = 7/14  =  1/2

 

So

 

arcsin ( 7/14)  =  arcsin (1/2)  =  30°  =  angle EAB 

 

So.....the  area of sector AEB  =   pi (r^2) (30/360)  =

pi ( 14)^2 (1/12)  =  (196/12) pi =  (49/3) pi  units^2    (2)

 

And the area of the  rectangle =  14 * 7  = 98 units^2     (3)

 

So....the shaded area =  (3) - (2) - (1)  =

 

98  - (49/2)sqrt (3) - (49/3) pi  ≈   4.252 units^2

 

 

cool cool cool

 Jan 8, 2020

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