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Warm Up Solve for x . 1. 2. 3 x = 12 2 3. BC and DC are tangent to  A . Find BC .

Warm Up Solve for x . 1. 2. 3 x = 12 2 3. BC and DC are tangent to  A . Find BC. 4. 48. 14. Objectives. Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems.

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Warm Up Solve for x . 1. 2. 3 x = 12 2 3. BC and DC are tangent to  A . Find BC .

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  1. Warm Up Solve for x. 1. 2. 3x = 122 3. BC and DC are tangent to A. Find BC. 4 48 14

  2. Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems.

  3. In 1901, divers near the Greek island of Antikythera discovered several fragments of ancient items. Using the mathematics of circles, scientists were able to calculate the diameters of the complete disks. The following theorem describes the relationship among the four segments that are formed when two chords intersect in the interior of a circle.

  4. J Example 1: Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. 10(7) = 14(x) 70 = 14x 5 = x EF = 10 + 7 = 17 GH = 14 + 5 = 19

  5. Check It Out! Example 1 Find the value of x and the length of each chord. 8(x) = 6(5) 8x = 30 x = 3.75 AB = 6 + 5 = 11 CD = 3.75 + 8 = 11.75

  6. Example 2: Art Application The art department is contracted to construct a wooden moon for a play. One of the artists creates a sketch of what it needs to look like by drawing a chord and its perpendicular bisector. Find the diameter of the circle used to draw the outer edge of the moon. 8(d – 8) = 9  9 Let d – 8 = x 8x = 81 x=81/8

  7. Example 2 Continued OR 8(d – 8) = 9  9 8d – 64 = 81 8d = 145

  8. 6 in. Archaeologists discovered a fragment of an ancient disk. To calculate its original diameter, they drew a chord AB and its perpendicular bisector PR. The chordthat the archeologists drew was 12 in. Find the disk’s diameter. 6(6) = 3(QR) 12 = QR 12 + 3 = 15 = PR

  9. A secant segmentis a segment of a secant with at least one endpoint on the circle. An external secant segmentis a secant segment that lies in the exterior of the circle with exactly one endpoint on the circle.

  10. Example 3: Applying the Secant-Secant Product Theorem Find the value of x and the length of each secant segment. 112 = 64 + 8x 48 = 8x 6 = x ED = 7 + 9 = 16 EG = 8 + 6 = 14

  11. Check It Out! Example 3 Find the value of z and the length of each secant segment. 351 = 169 + 13z 182 = 13z 14 = z LG = 30 + 9 = 39 JG = 14 + 13 = 27

  12. A tangent segmentis a segment of a tangent with one endpoint on the circle. ABand ACare tangent segments.

  13. Example 4: Applying the Secant-Tangent Product Theorem Find the value of x. ML JL = KL2 20(5) = x2 100 = x2 ±10 = x The value of x must be 10 since it represents a length.

  14. Check It Out! Example 4 Find the value of y. DE DF = DG2 7(7 + y) = 102 49 + 7y = 100 7y = 51

  15. Homework Page 795 Exercises 1-11

  16. Lesson Quiz: Part I 1. Find the value of d and the length of each chord. d = 9 ZV = 17 WY = 18 2. Find the diameter of the plate.

  17. Lesson Quiz: Part II 3. Find the value of x and the length of each secant segment. x = 10 QP = 8 QR = 12 4. Find the value of a. 8

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