11.2.3 Segment Relationships in Circles Key Objectives • Find the lengths of segments formed by lines that intersect circles. • Use the lengths of segments in circles to solve problems. Key Terms • A secant segment is a segment of a secant with at least one endpoint on the circle. • An external secant segment is a secant segment that lies in the exterior of the circle with one endpoint on the circle. • A tangent segment is a segment of a tangent with one endpoint on the circle. Theorems, Postulates, Corollaries, and Properties • Chord-Chord Product Theorem If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal. • Secant-Secant Product Theorem If two secants intersect in the exterior of a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment. (whole · outside = whole · outside) • Secant-Tangent Product Theorem If a secant and a tangent intersect in the exterior of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (whole ⋅ outside = tangent 2 ) Example 1 Applying the Chord-Chord Product Theorem
If two chords intersect in the interior of a circle, then the products of the lengths of the segments of the chords are equal.
The Chord-Chord Product Theorem is used in this example to determine the length of two chords. The lengths of three of the segments formed by two chords intersecting inside a circle are given. Substitute the given values into the equation for the Chord-Chord Product Theorem and solve for x . The solution yields x = 8. The lengths of the chords are BD = 14 and AC = 11.
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