A vertex of an equilateral triangle is (2,3) and the opposite side is x + y -2 =0 . Find the equations of other two sides .

Dear Student,

Please find below the solution to the asked query :
Line x + y = 2, i.e., y = (-1)x + 2, has slope m = -1. 

If M is the slope of a line which makes an angle of 60° 

with the above line, then : 

... tan 60° = | ( M - m ) / ( 1 + Mm ) | 

∴ √3 = | ( M + 1 ) / ( 1 - M ) | 

∴ squaring both sides and simplifying, 

... 3( 1 - M )² = ( M + 1 )² 

∴ 3 - 6M + 3M² = M² + 2M + 1 

∴ 2M² - 8M + 2 = 0 

∴ M² - 4M + 1 = 0 

∴ M = [ 4 ± √(16- 4) ] / 2 = 2 ± √3 

∴ the equations of the two sides, passing through (2,3), are 

... y - 3 = (2+√3)( x - 2 ) ... and ... y - 3 = (2-√3)( x - 2 )

Hope this would clear your doubt about the topic.

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