ABCD is a square of side a cm. AB, BC, CD and AD all are the chords of circles with equal radii each. If the chords subtends an angle of 120° at their respective centres, find the total area of the given figure where arcs are part of the circles:{http://silverzone.org/Syl_Sam_Papers_17-18/M/10.pdf --q no 8}
(A) 2 2 2 πa a a + 4 – 9 3 2             (B) 2 2 2 πa a a + 4 – 9 4 3             (C) 2 2   9a – 4π + 3 3a   (D) 2 2   9a + 4π – 3 3a    
 
This is a olympaid question so solve wisely but explain with steps

Asked by sucharitasahoo1 | 30th Nov, 2017, 09:04: PM

Expert Answer:

begin mathsize 16px style in space the space first space image space AB thin space is space not space the space diameter space it space is space straight a space chord
so space in space second space image space AOBX space is space sector
so space area space
Sector equals 120 over 360 cross times πr squared equals πr squared over 3
now space from space fig space 3 space
straight a over 2 equals straight r space sin space 60
straight r equals space fraction numerator straight a over denominator square root of 3 end fraction
Sector space AOBX equals πa squared over 9
now space the space area space of space triangle space increment AOB space open curly brackets space fig.3 close curly brackets
equals 1 half cross times AB cross times OS
equals 1 half cross times straight a cross times open parentheses fraction numerator straight a over 2 over denominator space tan 60 end fraction close parentheses....... open curly brackets AB equals 2 straight a comma OS equals fraction numerator straight a over 2 over denominator space tan 60 end fraction space close curly brackets
space increment AOB equals fraction numerator straight a squared over denominator 4 square root of 3 end fraction
now space total space area space of space figure space left parenthesis 1 right parenthesis
equals area space of space square space square ABCD plus open parentheses area space of space sector space AOBX close parentheses cross times 4 minus open parentheses area space of space space increment AOB close parentheses cross times 4
equals straight a squared plus πa squared over 9 cross times 4 minus fraction numerator straight a squared over denominator 4 square root of 3 end fraction cross times 4 end style

Answered by Arun | 1st Dec, 2017, 03:54: PM