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Intersecting Secant-Tangent Theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

Math diagram

In the circle, $\bar{U V}$ is a tangent and $\bar{U Y}$ is a secant. They intersect at point $U$ .

So, $U V^{2} = U X \cdot U Y$ .

Example :

In the circle shown, if $U X = 8$ and $X Y = 10$ , then find the length of $U V$ .

Substitute.

$\begin{array}{l} U V^{2} = U X \cdot U Y \\ = (U X) (U X + X Y) \\ = (8) (8 + 10) \\ = (8) (18) \\ = 144 \end{array}$

Take the square root on each side.

$\begin{array}{l} U V^{2} = \sqrt{144} \\ U V = 12 \end{array}$