12 6 Segment Relationshipsinin Circles Warm Up Lesson

12 -6 Segment. Relationshipsinin. Circles Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Geometry Holt

12 -6 Segment Relationships in Circles Warm Up Solve for x. 1. 4 2. 3 x = 122 48 3. BC and DC are tangent to A. Find BC. 14 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Objectives Find the lengths of segments formed by lines that intersect circles. Use the lengths of segments in circles to solve problems. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Vocabulary secant segment external secant segment tangent segment Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles 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. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Example 1: Applying the Chord-Chord Product Theorem Find the value of x and the length of each chord. EJ JF = GJ JH 10(7) = 14(x) 70 = 14 x 5=x EF = 10 + 7 = 17 GH = 14 + 5 = 19 Holt Mc. Dougal Geometry J

12 -6 Segment Relationships in Circles Check It Out! Example 1 Find the value of x and the length of each chord. DE EC = AE EB 8(x) = 6(5) 8 x = 30 x = 3. 75 AB = 6 + 5 = 11 CD = 3. 75 + 8 = 11. 75 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles 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. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Example 2 Continued 8 (d – 8) = 9 9 8 d – 64 = 81 8 d = 145 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Check It Out! Example 2 What if…? Suppose the length of chord AB that the archeologists drew was 12 in. In this case how much longer is the disk’s diameter compared to the disk on p. 793? AQ QB = PQ QR 6(6) = 3(QR) 12 = QR 12 + 3 = 15 = PR Holt Mc. Dougal Geometry 6 in.

12 -6 Segment Relationships in Circles 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. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Example 3: Applying the Secant-Secant Product Theorem Find the value of x and the length of each secant segment. 16(7) = (8 + x)8 112 = 64 + 8 x 48 = 8 x 6=x ED = 7 + 9 = 16 EG = 8 + 6 = 14 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Check It Out! Example 3 Find the value of z and the length of each secant segment. 39(9) = (13 + z)13 351 = 169 + 13 z 182 = 13 z 14 = z LG = 30 + 9 = 39 JG = 14 + 13 = 27 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles A tangent segment is a segment of a tangent with one endpoint on the circle. AB and AC are tangent segments. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Example 4: Applying the Secant-Tangent Product Theorem Find the value of x. ML JL = KL 2 20(5) = x 2 100 = x 2 ± 10 = x The value of x must be 10 since it represents a length. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles Check It Out! Example 4 Find the value of y. DE DF = DG 2 7(7 + y) = 102 49 + 7 y = 100 7 y = 51 Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles 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. Holt Mc. Dougal Geometry

12 -6 Segment Relationships in Circles 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 Holt Mc. Dougal Geometry