1. Prove that two parallel secants intercept equal arcs on a circle.

1. § 10.1 C D 1. Prove that two parallel secants intercept equal arcs on a circle. Q O A B P Given: AB  CD. Prove: arc AC = arc BD G F E

2. A O B C 2. Prove that an inscribed angle, with the center of the circle on the exterior of the angle, is half the measure of the intercepted arc. D Given: Inscribed angle ABC with O on the exterior of the angle. Prove: ABC= ½ arc AC O

3. A O B C 2. Prove that an inscribed angle, with the center of the circle on the interior of the angle, is half the measure of the intercepted arc. D Given: Inscribed angle ABC with O on the interior of the angle. Prove: ABC= ½ arc AC

4. 3. Prove that an inscribed angle, with the center of the circle on the exterior of the angle, is half the measure of the intercepted arc. • KN = 6 tangents from the same point (K) are equal. • KO = 6 + 4 = 10 • JP = 6. • LN = 3 a radius • LO = 5 from Pythagorean theorem on triangle LNO a 3-4-5 triangle • PO = LO – LP = 5 – 3 = 2 • KL = √45 from Pythagorean theorem on triangle LNK a 3-6-x triangle

5. 4. Prove that the angle formed by a secant and a tangent is half the difference of the intercepted arcs. A D Given: Secant AB and tangent BC Prove: B = ½ arc (AC – CD). B E C

6. D E 5. Prove that the angle formed by two tangents is half the difference of the intercepted arcs. A Given: Tangent AB and tangent BC Prove: B = ½ arc (ADC – AC). C B

7. 6. The sum of the lengths of two tangent segments to a circle form the same exterior point is equal to the diameter of the circle. Find the measure of the angle determined by the tangent segments B DC = CE and DC + CE = AB making each of the tangent segments a length equal to the radius of the circle. This means that ODCE would be a rhombus but since the radii are perpendicular to the tangent segments it is a square and hence the angle at C is 90. E A C D

8. 7. If point P is 13 inches from the center of a circle with a 10 inch diameter, how long are the tangent segments from P? B 5 O P 5 13 A 5 Use the Pythagorean Theorem on triangle OBP with leg of 5 and hypotenuse of 13 to find the length of the other leg or tangent segment. C

9. 8. Two chords of a circle intersect. The segments of one chord have lengths 4 and 6. if the length of one segment of the other chord is 3, find the length of the remaining segment. C X 4 A 6 Use the two-chord power theorem. 3x = (4)(6) and x = 8. 3 B D

10. 9. Solve for x. Use the theorem that x is half the sum of the intercepted arcs.

11. 10. Solve for x. Use the theorem that x is half the difference of the intercepted arcs.

12. 11. Solve for x. Use the theorem that x is half the difference of the intercepted arcs and the fact that the missing measurement is 360 – 92 or 268.

13. 12. Find QR  QS when: • a. QS = 9 and QR = 5 • b. QU = 14 and QT = 10 • c. QT = 1 and TU = 13 • d. QR = 2 and SR = 7 • Use the two-chord power theorem: QR  QS = QU  QT • QR  QS = 9  5 = 45. • QR  QS = QU  QT = 14  10 = 140 • QR  QS = QU  QT = 12  1 = 12 • QR  QS = 2  5 = 10

14. 13. Segment DB is a diameter of a circle. A tangent through D and a secant through B intersect at a point A. The secant also intersects the circle at C. Prove the DB 2 = (AB)(BC). D From the tangent-secant theorem we know that AB  AC = AD 2 AB  (AB – BC) = AD 2 Since AC = AB – BC AB 2 – AB  BC = AD 2 Distributive property of multiplication. AB 2 – AD 2 = AB  BC Rearrangement of equation. BD 2 = AB  BC Since BD 2 = AB 2 – AD 2 by the Pythagorean Theorem. O A C B