10 6 Find Segment Lengths in Circles Hubarth
10 -6 Find Segment Lengths in Circles Hubarth Geometry
Theorem 10. 14 Segments of Chords Theorem 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 chords. C A E D B
Ex 1 Find Segments Lengths Find ML and JK. NK NJ = NL NM x (x + 4) = (x + 1) (x + 2) x 2 + 4 x = x 2 + 3 x + 2 4 x = 3 x + 2 x = 2 ML = ( x + 2 ) + ( x + 1) JK = x + ( x + 4) = 2+2+2+1 = 2+2+4 = 7 = 8
Theorem 10. 15 Segments of Secant Theorem 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. B A E C D
EX 2 Standardized Test Practice RQ RP 4 (5 + 4) = RS RT = 3 (x + 3) 36 = 3 x + 9 9 = x The correct answer is D.
Theorem 10. 16 Segments of Secant and Tangents Theorem If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment. A E C D
Ex 3 Find Lengths Using Theorem 10. 16 Use the figure at the right to find RS. RQ 2 162 256 = RS RT = x (x + 8) = x 2 + 8 x 0 = x 2 + 8 x – 256 x – 8 + = 82 – 4(1) (– 256) 2(1) x= – 4+ 4 17
Ex 4 Solve a Real-World Problem Tethys, Calypso, and Telesto are three of Saturn’s moons. Each has a nearly circular orbit 295, 000 kilometers in radius. The Cassini-Huygens spacecraft entered Saturn’s orbit in July 2004. Telesto is on a point of tangency. Find the distance DB from Cassini to Tethys. DC DB = AD 2 203, 0002 83, 000 DB DB 496, 494 Cassini is about 496, 494 kilometers from Tethys.
Practice Find the value(s) of x. 13 = x 4. Find the value of x. x=2 x = 8 3 = x 5. Find the value of x. x= 24 5
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