How do you find x when line AB, CD, and AD are tangent of circle O? (Point O is the centre of the circle, line BO and OC are radius of circle O.)[**NOT DRAWN ACCURATELY**]

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1 Answer
CW
Oct 21, 2017

#x=2.1# units

Explanation:

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Given that #B,C and M# are the points of tangency.
As the two tangent segments to a circle from an external point are equal, #=> AM=AB=3, MD=CD=7#,
#BC=AE=sqrt(AD^2-DE^2)=sqrt(10^2-4^2)=sqrt84=2sqrt21#
As #AB# is parallel to #CD, => DeltaANB and DeltaCND# are similar,
#=> (AN)/(CN)=(AB)/(CD)=3/7#
As #(AM)/(DM)=3/7, => MN# is parallel to #AB and CD#,
extend #MN# to #P#, #=> (BP)/3=(AE)/10, => BP=(3sqrt21)/5#
#(NP)/(BP)=(DC)/(BC), => NP=(3sqrt21)/5*7/(2sqrt21)=21/10#
#MP=3+(3sqrt21)/5*4/(2sqrt21)=21/5#
#=> x=MN=MP-NP=21/5-21/10=21/10=2.1#

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