Intersecting Tangent Secant Theorem

What is the Secant-Tangent Rule?

If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment.

The following figures give the set operations and Venn Diagrams for complement, subset, intersect and union. Scroll down the page for more examples and solutions.

Statement of the Secant-Tangent Theorem
If a tangent and a secant are drawn from an external point to a circle, then:
(Length of tangent)² = (Length of secant) × (Length of external segment of secant).

Mathematically, if:
PT is the length of the tangent,
PA is the length of the secant from the external point to the first intersection with the circle,
PB is the length of the secant from the external point to the second intersection with the circle,
then:
PT² = PA × PB.

Segments of Secants and Tangents Theorem
The segments of a secant segment and a tangent segment which share an endpoint outside of the circle. The product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Power of a point with a tangent segment and a secant segment

Tangent - secant demonstration
Demonstration of a circle geometry property

Tangent Secant Theorem (with quadratic expressions) - Geometry
This video focuses on using the Tangent Secant Theorem to find the length of a tangent line segment. This video also reviews how to find the roots of a quadratic equation.

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