Circles and Chords Topic Index | Geometry Index | Regents Exam Prep Center

 A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle.

 Theorems:
 1. 2. 3. In a circle, a radius perpendicular to a chord bisects the chord. In a circle, a radius that bisects a chord is perpendicular to the chord. In a circle, the perpendicular bisector of a chord passes through the center of the circle.

Proof of Theorem 1:

 Statements Reasons 1. 1. Given 2. 2. Two points determine exactly one line. 3. 3. Perpendicular lines meet to form right angles. 4. 4. A right triangle contains one right angle. 5. 5. Radii in a circle are congruent. 6. 6. Reflexive Property - A segment is congruent to itself. 7. 7. HL - If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. 8. 8. CPCTC - Corresponding parts of congruent triangles are congruent. 9. E is the midpoint of 9. Midpoint of a line segment is the point on that line segment that divides the segment two congruent segments. 10. 10. Bisector of a line segment is any line (or subset of a line) that intersects the segment at its midpoint.

 Theorem:
 In a circle, or congruent circles, congruent chords are equidistant from the center. (converse) In a circle, or congruent circles, chords equidistant from the center are congruent.

 Theorem:
 In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords. ;

 Theorem:
 In a circle, parallel chords intercept congruent arcs.

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