SOLUTION: Let B, C, and D be points on a circle. Let BC and the tangent to the circle at D intersect at A. If AB = 4, AD = 8, and AC is perpendicular to AD, then find CD. https://latex.ar

Question 1074109: Let B, C, and D be points on a circle. Let BC and the tangent to the circle at D intersect at A. If AB = 4, AD = 8, and AC is perpendicular to AD, then find CD.
https://latex.artofproblemsolving.com/5/4/7/547ab841560cd28dd3d03fb5510dd297b29e8a42.png
Answer by ikleyn(52754) (Show Source): You can put this solution on YOUR website!
.

The key to the solution of the problem is this theorem


     Theorem

     If a tangent and a secant lines are released from a point outside a circle, 
     then the product of the measures of the secant and its external part is equal to the square of the tangent segment.


For the proof, see the lesson

    - Metric relations for a tangent and a secant lines released from a point outside a circle

in this site.


Using the theorem, you can find the length of the secant AC:  |AC| =  =  =  = 16 units.


Then |CD| =  =  = .

Solved.

Also, you have this free of charge online textbook on Geometry
GEOMETRY - YOUR ONLINE TEXTBOOK
in this site.

The referred lessons are the part if this textbook under the topic
"Properties of circles, inscribed angles, chords, secants and tangents".

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