In figure ABCD is a rhombus whose diagonal AC makes an angle alpha with AB. If cos alpha = 2/3 & OB = 3cm, find the lengths of diagonals AC & BD.

Asked by araima2001 | 1st Sep, 2015, 05:48: PM

Expert Answer:

begin mathsize 14px style in space increment AOB angle straight O space equals space 90 space open parentheses Diagonals space of space rhombus space AC perpendicular BD close parentheses OB space equals space 3 space cm space open parentheses Given close parentheses space space space space space... open parentheses 1 close parentheses cos space straight alpha space equals space 2 over 3 space space therefore AO over AB equals 2 over 3 therefore AO equals 2 over 3 AB space space space... open parentheses 2 close parentheses open parentheses AB close parentheses squared equals open parentheses AO close parentheses squared plus open parentheses OB close parentheses squared therefore open parentheses AB close parentheses squared equals open parentheses 2 over 3 AB close parentheses squared plus 3 squared space space space space space space space space... open parentheses 1 close parentheses space space and space space... open parentheses 2 close parentheses therefore open parentheses AB close parentheses squared equals 4 over 9 AB squared plus 9 space rightwards double arrow open parentheses AB close parentheses squared equals fraction numerator 4 AB squared plus 81 over denominator 9 end fraction therefore 9 open parentheses AB close parentheses squared equals 4 AB squared plus 81 space rightwards double arrow 5 open parentheses AB close parentheses squared equals 81 space therefore open parentheses AB close parentheses squared equals 81 over 5 rightwards double arrow AB space equals space fraction numerator 9 over denominator square root of 5 end fraction Substituting space open parentheses AB close parentheses squared equals 81 over 5 space in space open parentheses AB close parentheses squared equals open parentheses AO close parentheses squared plus open parentheses OB close parentheses squared 81 over 5 equals open parentheses AO close parentheses squared plus 9 space rightwards double arrow open parentheses AO close parentheses squared equals 81 over 5 minus 9 space therefore space open parentheses AO close parentheses squared equals 81 over 5 minus 9 equals space fraction numerator 81 minus 45 over denominator 5 end fraction therefore space open parentheses AO close parentheses squared equals 36 over 5 AO equals space fraction numerator 6 over denominator square root of 5 end fraction Diagonal space AC space equals space 2 space open parentheses AO close parentheses space equals space 2 cross times fraction numerator 6 over denominator square root of 5 end fraction equals fraction numerator 12 over denominator square root of 5 end fraction cm open parentheses because space Diagonals space of space rhombus space bisect space each space other close parentheses  Diagonal space BD space equals space 2 space open parentheses OB close parentheses space equals space 2 cross times 3 equals 6 space cm                    end style

Answered by Priti Shah | 1st Sep, 2015, 07:21: PM

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