AB is a chord of circle E, and C is the midpoint of AB. Prove that EC is the perpendicular bisec

step 1 Question number 14 here is given that AB is the card of a circle E and C is the midpoint of the

AB is a chord of circle E, and C is the midpoint of AB....

$\overline{\mathrm{AB}}$ is a chord of circle $\mathrm{E},$ and $\mathrm{C}$ is the midpoint of $\overline{\mathrm{AB}}$. Prove that $\overrightarrow{\mathrm{EC}}$ is the perpendicular bisector of chord $\overline{\mathrm{AB}}$.

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Key Concepts

Circle

A circle is defined as the set of all points in a plane that are at a fixed distance (the radius) from a central point. This central symmetry results in unique properties, such as the fact that any perpendicular bisector of a chord in the circle will pass through the center, thus emphasizing the circle's symmetrical nature.

Chord

A chord is a line segment with both endpoints on the circle. Its properties are closely linked to symmetry; for example, chords equidistant from the center are equal in length, and the perpendicular bisector of any chord will pass through the circle’s center, highlighting a key geometric relationship.

Midpoint

The midpoint of a segment is the point that divides the segment into two equal parts. In the context of a chord in a circle, the midpoint serves as the point of symmetry, and a line drawn from the center through this midpoint will bisect the chord perpendicularly.

Perpendicular Bisector

A perpendicular bisector is a line that divides a segment into two equal parts at a right angle. In circle geometry, the perpendicular bisector of any chord is significant because it always passes through the circle’s center, thus confirming and utilizing the inherent symmetry of the circle.

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