Applying Properties of Chords

Circumcenter—the point that is equidistant from each vertex of the triangle. It is the “center” of the triangle.

circumcenter of an equilateral triangle

Chord Bisector Theorem— The perpendicular bisector of a chord passes through the center of the circle (It would be a diameter).

CD is perpendicular to AB and DE = CE, then AB is a diameter. AB passes through the center of the circle A diameter that is perpendicular to a chord bisects the chord and its corresponding arc.

If CD is perpendicular to AB and AB is a diameter (which passes through the center point), then CE = DE and m arc CB = m arc DC

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.

 AB is congruent to CD if and only if they are an equal distance away from P EP is congruent to FP AB = 34 CD = 34 Two ways to show two chords are congruent: 1. Show that their corresponding arcs are congruent 2. Show that they are equidistant from the center of the circle. In a circle, a radius perpendicular to a chord bisects the chord. Proof of Theorem 1:

Statements Reasons 1. 1. Given

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