Section 11-2 Chords and Arcs SPI 32B: Identify chords of circles given a diagram SPI 33A: Solve problems involving the properties of arcs, tangents, chords.

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Presentation transcript:

Section 11-2 Chords and Arcs SPI 32B: Identify chords of circles given a diagram SPI 33A: Solve problems involving the properties of arcs, tangents, chords Objectives: Use congruent chords, arcs and central angles Chord of a circle A segment whose endpoints are on a circle A B Arc Central Angle

Relate Central Angles, Chords and Arcs Theorem 11-4: Within a circle or in congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles. In the diagram, radius OX bisects AOB. What can you conclude? AOX BOX by the definition of an angle bisector. AX BX because congruent central angles have congruent chords. AX BX because congruent chords have congruent arcs.

Relate Central Angles, Chords and Arcs Theorem 11-5: Within a circle or in congruent circles: (1) Chords equidistant from the center are congruent. (2) Congruent chords are equidistant from the center QS = QR + RSSegment Addition Postulate QS = 7 + 7Substitute. QS = 14Simplify. AB = QSChords that are equidistant from the center of a circle are congruent. AB = 14Substitute 14 for QS. Find AB.

Lines Through the Center of a Circle Theorem 11-6: In a circle, a diameter that is perpendicular to a chord bisects the chord and its arcs. Theorem 11-7: In a circle, a diameter that bisects a chord (that is NOT a diameter) is perpendicular to the chord. Theorem 11-8: In a circle, the perpendicular bisector of a chord contains the center of the circle.

OP 2 = PM 2 + OM 2 Use the Pythagorean Theorem. r 2 = Substitute. r 2 = 289Simplify. r = 17Find the square root of each side. PM = PQA diameter that is perpendicular to a chord bisects the chord PM = (16) = 8Substitute The radius of O is 17 in.. Draw a diagram to represent the situation. The distance from the center of O to PQ is measured along a perpendicular line.. P and Q are points on O. The distance from O to PQ is 15 in., and PQ = 16 in. Find the radius of O. (Hint: Draw a diagram).. Relate Central Angles, Chords and Arcs

Find each missing length to the nearest tenth.