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
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