10-6 Find Segment Lengths in Circles

Segments of Chords Theorem m n p m n = p q If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. q

example: solve for x = 6 x 24 = 6x x = x

Definitions Tangent Segment – a piece of a tangent with one endpoint at the point of tangency. Secant Segment – a piece of a secant containing a chord, with one endpoint in the exterior of the circle and the other on the circle. External Secant Segment – the piece of a secant segment that is outside the circle. S P SP R RP Q PQ

Segments of Secants Theorem C B A If two secant segments share the same endpoint outside 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. D AB AC = AD AE E

example: solve for x 11 * 21 = 12 (12 + x) 231 = x 231 = x 87 = 12x 87 = 12x 7.25 = x 7.25 = x x 12 x 12 X = 21

Segments of Secants and Tangents Theorem A B D If two secant segments share the same endpoint outside 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. C (AB) 2 = AC AD

example: solve for x 30 2 = x (x + 24) 30 2 = x (x + 24) 900 = x x 900 = x x x 2 +24x – 900 = 0 x 2 +24x – 900 = 0 How do you solve for x? Use the quadratic formula!! x = x = x 30 a = 1 b = 24 c = -900

example: solve for x x ̊ 140 ̊ 1 st : Find the other arc 360 – 140 = – 140 = x 2 80 = x 2 40 ̊ = x 220 ̊