Segment Relationships in Circles 10-5 Segment Relationships in Circles Warm Up Lesson Presentation Lesson Quiz Holt Geometry

Warm Up Solve for x. 1. 2. 3x = 122 4 3. BC and DC are tangent 10.5 Segment Relationships in Circles Warm Up Solve for x. 1. 2. 3x = 122 3. BC and DC are tangent to A. Find BC. 4 48 14

10.5 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.

Vocabulary secant segment external secant segment tangent segment 10.5 Segment Relationships in Circles Vocabulary secant segment external secant segment tangent segment

10.5 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.

10.5 Segment Relationships in Circles

Example 1: Applying the Chord-Chord Product Theorem 10.5 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 J 10(7) = 14(x) 70 = 14x 5 = x EF = 10 + 7 = 17 GH = 14 + 5 = 19

10.5 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) 8x = 30 x = 3.75 AB = 6 + 5 = 11 CD = 3.75 + 8 = 11.75

Example 2: Art Application 10.5 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.

10.5 Segment Relationships in Circles Example 2 Continued 8  (d – 8) = 9  9 8d – 64 = 81 8d = 145

10.5 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 in. 6(6) = 3(QR) 12 = QR 12 + 3 = 15 = PR

10.5 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.

10.5 Segment Relationships in Circles

Example 3: Applying the Secant-Secant Product Theorem 10.5 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 + 8x 48 = 8x 6 = x ED = 7 + 9 = 16 EG = 8 + 6 = 14

10.5 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 + 13z 182 = 13z 14 = z LG = 30 + 9 = 39 JG = 14 + 13 = 27

10.5 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.

10.5 Segment Relationships in Circles

Example 4: Applying the Secant-Tangent Product Theorem 10.5 Segment Relationships in Circles 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.

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

10.5 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.

8 10.5 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

10.5 Segment Relationships in Circles Video 1. Find the value of x and the length of each secant segment.