Prove the stated theorem. If a line is drawn through the...

Prove the stated theorem.
If a line is drawn through the center of a circle perpendicular to a chord, then it bisects the chord and its minor arc. See Figure 6.37 on page 288 (NOTE: The major arc is also bisected by the line.)

Key Concepts

Central Symmetry

Central symmetry is a property of circles where every line through the center divides the circle into two congruent halves. This symmetry ensures that elements like chords and arcs are split into equal parts when intersected by a line through the center, providing the basis for many proofs in circle geometry.

Circle

A circle is defined as the set of all points in a plane that are equidistant from a fixed point known as the center. This property gives rise to many symmetrical characteristics, which are fundamental in many geometric proofs and theorems.

Chord

A chord is a line segment with both endpoints on the circle’s circumference. Its properties are closely tied to the circle's symmetry, notably that its perpendicular bisector always passes through the center of the circle.

Perpendicular Bisector

A perpendicular bisector of a segment is a line that divides the segment into two equal parts at a 90-degree angle. In circle geometry, a line drawn through the center perpendicular to a chord not only bisects the chord but also the corresponding arc, due to the inherent symmetry of the circle.

Triangle Congruence

Triangle congruence is a fundamental concept where two triangles are considered equal in shape and size if their corresponding sides and angles are congruent. This concept is often used in proofs within circle geometry to establish equal lengths and angles, supporting the bisecting properties of chords and arcs.

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