SOLUTION: A secant and a tangent segment intersect in the exterior of a circle. A chord which is a part of the secant is six more than the length of the tangent segment and the external se

Algebra ->  Circles -> SOLUTION: A secant and a tangent segment intersect in the exterior of a circle. A chord which is a part of the secant is six more than the length of the tangent segment and the external se      Log On


   

Question 1175440: A secant and a tangent segment intersect in the exterior of a circle. A chord
which is a part of the secant is six more than the length of the tangent
segment and the external segment of the secant is 6cm. Find the length of the
tangent segment.
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The operative fact here is that the square of the length of the tangent is equal to the product of the length of the entire secant and the length of the external part of the secant.

In your problem, the length of the tangent is x, the length of the internal part of the secant is x+6, and the length of the external part of the secant is 6. That makes the length of the entire secant x+12. So

x%5E2+=+6%28x%2B12%29
x%5E2+=+6x-72
x%5E2-6x-72+=+0
%28x-12%29%28x%2B6%29+=+0
x = 12 or x = -6. Since x is the length of the tangent segment, reject the negative solution.

ANSWER: The length of the tangent segment is x=12cm.

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NOTE: If a similar problem is about two secants, then the product of the length of the external part and the entire length of one secant is equal to the product of the length of the external part and the entire length of the other secant.