http://www.ck12.org Chapter 9. Circles
9.11 Segments from Secants and Tangents
Here you’ll learn how to solve for missing segments created by a tangent line and a secant line intersecting outside
a circle.
What if you were given a circle with a tangent and a secant that intersect outside the circle? How could you use
the length of some of the segments formed by their intersection to determine the lengths of the unknown segments?
After completing this Concept, you’ll be able to use the Tangent Secant Segment Theorem to solve problems like
this one.
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If a tangent and secant meet at a common point outside a circle, the segments created have a similar relationship to
that of two secant rays. Recall that the product of the outer portion of a secant and the whole is equal to the same of
the other secant. If one of these segments is a tangent, it will still be the product of the outer portion and the whole.
However, for a tangent line, the outer portion and the whole are equal.
Tangent Secant Segment Theorem:If a tangent and a secant are drawn from a common point outside the circle
(and the segments are labeled like the picture to the left), thena^2 =b(b+c). This means that the product of the
outside segment of the secant and the whole is equal to the square of the tangent segment.
Example A
Find the value of the missing segment.
Use the Tangent Secant Segment Theorem. Square the tangent and set it equal to the outer part times the whole
secant.