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Lesson 10.1

Lesson 10.1. Circles. Definition:. The set of all points in a plane that are a given distance from a given point in the plane. The given point is the CENTER of the circle. A segment that joins the center to a point on the circle is called a radius.

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Lesson 10.1

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  1. Lesson 10.1 Circles

  2. Definition: • The set of all points in a plane that are a given distance from a given point in the plane. • The given point is the CENTER of the circle. • A segment that joins the center to a point on the circle is called a radius. • Two circles are congruent if they have congruent radii.

  3. Concentric Circles:Two or more coplanar circles with the same center.

  4. A point is inside (in the interior of) a circle if its distance from the center is less than the radius. interior A Point O and A are in the interior of Circle O. O

  5. A point is outside (in the exterior of) a circle if its distance from the center is greater than the radius. W A Point W is in the exterior of Circle A. S A point is on a circle if its distance from the center is equal to the radius. Point S is on Circle A.

  6. Chords and Diameters: • Points on a circle can be connected by segments called chords. • A chord of a circle is a segment joining any two points on the circle. • A diameter of a circle is a chord that passes through the center of the circle. • The longest chord of a circle is the diameter. chord diameter

  7. Formulas to know! Circumference: • C = 2 π r or • C = πd • Area: • A = πr2

  8. Radius-Chord Relationships • OP is the distance from O to chord AB. • The distance from the center of a circle to a chord is the measure of the perpendicular segment from the center to the chord.

  9. Theorem 74 If a radius is perpendicular to a chord, then it bisects the chord.

  10. Theorem 75 If a radius of a circle bisects a chord that is not a diameter, then it is perpendicular to that chord.

  11. Theorem 76 The perpendicular bisector of a chord passes through the center of the circle.

  12. 1. Circle Q, PR  ST • PR bisects ST. • PR is  bisector of ST. • PS  PT • Given • If a radius is  to a chord, it bisects the chord. (QR is part of a radius.) • Combination of steps 1 & 2. • If a point is on the  bisector of a segment, it is equidistant from the endpoints.

  13. The radius of Circle O is 13 mm.The length of chord PQ is 10 mm. Find the distance from chord PQ to center, O. Draw OR perpendicular to PQ. Draw radius OP to complete a right Δ. Since a radius perpendicular to a chord bisects the chord, PR = ½ PQ = ½ (10) = 5. By the Pythagorean Theorem, x2 + 52 = 132 The distance from chord PQ to center O is 12 mm.

  14. ΔABC is isosceles (AB  AC) Circles P & Q, BC ║ PQ ABC P, ACB Q ABC ACB P  Q AP  AQ PB  CQ Circle P  Circle Q • Given • Given • ║Lines means corresponding s . • . • Transitive Property • . • Subtraction (1 from 6) • Circles with  radii are .

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