Circles and Chords
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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.


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Theorem:

In a circle, parallel chords intercept congruent arcs.