1. CIRCLES Kelompok 6 Asti Pujiningtyas 4101414009 Eva Wulansari 4101414023 Mifta Zuliyanti4101414016 Zuliyana Dewi A. 4101414001

2. 10-1 Basic Definitions Definition 10-1 A radius of a circle is a segment whose endpoints are the center and a point on the circle. A B Radius A D C Definiton 10-2 A chord of a circle is a segment with endpoints on the circle. chord

3. A Definition 10-3 A diameter of a circle is a chord that contain the center of the circle. Definition 10-4 A tangent to a circle is a line that intersect the circle in exactly one point. Definition 10-5 A secant of a circle is a line that intersect the circle in exactly two points. GH l A B Diameter A D E m

4. Definition 10-6 An inscribed angle is an angle with vertex on a circle and with sides that contain chords of the circle. Definition 10-7 A central angle is an angle with vertex at the center of a circle. A H I G A K J

5. 10-2 The Degree Measure of Arcs Definition 10-8 A minor arc is an arc that lies in the interior of a central angle. Otherwise ut is called a major arc Definition 10-9 The measure of a minor arc is the measure of its associated central angle. The measure of a major arc is 360 minus the measure of its associated minor arc. O A B Minor arc Major arc AA B 70 Arc Addition Postulate If C is on AB, then mAC + mCB = mAB

6. Definition 10-10 If two arcs of a circle have the same measure, they are called congruent. If AB and CD are congruent, we write AB CD. Definition 10-11 Two circles are congruent if they have radii of equal lenght. A B D C 50° D C B A

7. These two figures should focus your attention on the relationship between congruent chords and their arcs Given congruent chords Given congruent AB CD A B C D A B C D

8. A B C D Theorem 10-1 In a circle or in congruent circles congruent chords have congruent minor arcs. O

9. Theorem 10-2 In a circle or in congruent circles congruent minor arcs have congruent chords. A B C D StatementReason 1. AB CDGiven 2. OA=OB=OC=ODDefinitoin of Circle 3.Definition of congruent segment 4.SAS Postulate 5.CPCTC O

10. In each figure a pair of congruent chords is given. In each case does XL = XM? These examples suggest the following theorem. 10-3. Chords and Distances from the Center

11. Theorem 10-3. In a circle or in congruent circles congruent chords are equidistant from the center. PROOF Given : circle O,,, Prove : OM = OL

12. StatementsReasons 1.1. Given 2. OA = OB = OC = OD2. Definition of circle 3.3. Definition of congruent segments 4.4. SSS congruence 5.5. CPCTC 6. and6. Given 7., and are right angles. 7. Perpendicular lines from congruent right angles 8. and are right triangles 8. Definition of right triangles 9.9. HA Congruence 10.10. CPCTC 11. OM = OL11. Definition of congruent segments

13. Theorem 10-4. In a circle or in congruent circles chords equidistant from the center are congruent PROOF Given : ʘ O, OM = OL, and Prove :

16. Theorem 10.5 The perpendicular bisector of a chord contain the center of the circle PROOF: Given: is a chord of circle O, and l is the perpendicular bisector of Prove: O is a point of l O B A l

17. StatementReason 1. l is the perpendicular bisector of 2.OA = OB 3.O lies on l 1.Given 2.Definition of circle 3.A point equidistant from point A and B belongs to the perpendicular bisector of (Theorem 6-10) O B A l

18. APPLICATION Find the center of around table. Step 1 Select any two chords, and Step 2 Draw the perpendicular bisector p of, and perpendicular bisector q of. Conclusion: By the Theorem 10-5 the center lies on both lines p and q. Consequently, the center of the table must be the intersection of these lines. O A B p D C q

19. Theorem 10.6 If a line through the center of a circle is perpendicular to a chord that is not diameter, then it bisects the chord and its minor arc. O A B C

20. StatementReasons 1. 2.OB = OA 3. 4. 5.∆ OCB ∆ OCA 6. 7.BC = CA 8. 9.AC BC 1. Given 2.Definition of Circle 3.Definition of congruent segment 4.Reflective property 5.HL Theorem 6.CPCTC 7.Definition of congruent segments 8.CPCTC 9.Definition 10-10 O A B C

21. Theorem 10.7 If a line through the center of a circle bisects a chord that is not a diameter, then its perpendicular to the chord O A B C

22. StatementReasons 1. 2. 3. 4.∆ OCB ∆ OCA 5. 6. 1.Given 2.Definition of Circle 3.Reflective property 4.SSS Postulate 5.CPCTC 6.Perpendicular lines from congruent right angles O A B C

23. 10-5 Tangents to Circles A line is tangent to a circle if it intersects the circle in exactly one point. A ℓ. O

24. A ℓ. O ℓ A. O B

25. A ℓ. O Statements 4 and 5 are contradictory. Hence the assumption is false and the line ℓ is tangent to the circle. ℓ A. O B ℓ A. O B

26. . O C.. B A A. O D EB.. C

27. intersect the circle at two different points, so is not tangential to the circle. Hence the assumption is false and the radius is perpendicular to tangent.

28. PROOF Given : is a tangent of circle O and ℓ is the perpendicular of. Prove : O is a point of ℓ.. C O. B A. ℓ

29. StatementReason 1.ℓGiven 2.ℓ does not contain point OIndirect proof assumption 3.Draw radius from O to point C Construction 4. Theorem 10-9 There is exactly one line through C that perpendicular to, so line ℓ contain the center of the circle