Geometry Circles

Segments in Circles

Segments refer to the regions defined by two endpoints on a circle and the line connecting them.

What is a circle?

A circle is the set of all points in a plane that are equidistant from a specific point known as the center.

Definition of a circle

The mathematical definition states that a circle is constituted by all points lying at a fixed distance (the radius) from the center point.

Circle P notation

A circle with center at point P is denoted as ⊙P.

Radius

A radius is the line segment connecting the center of the circle to any point on the circle, all radii of a circle are equal.

Chord

A chord is a segment whose endpoints are both located on the circumference of the circle.

Diameter

A diameter is a specific type of chord that passes through the center of the circle, effectively dividing it into two equal halves.

Tangent

A tangent is a line (or segment or ray) that touches the circle at exactly one point, referred to as the point of tangency.

Secant

A secant is a line that intersects the circle at two distinct points, effectively slicing through the circle.

Identification of segments in a circle

When labeling segments, lines, or points in ⊙P, it is essential to specify the names of each segment accurately.

Coplanar Circles and Common Tangents

Coplanar circles are circles that lie in the same plane, and they can have common tangents, which are lines touching both circles.

Properties of Tangents

A line being tangent to a circle means it is perpendicular to a radius of the circle at the point where it meets the circle.

Congruent Tangent Segments

Tangent segments drawn from a common external point are equal in length.

True or False Statements about Circles

Determine the validity of statements concerning circles, such as whether circles with the same radius are congruent.

Congruent vs Similar Circles

Circles are considered congruent if they possess the same radius; all circles are inherently similar due to their identical shape.

Understanding Inscribed Angles

An inscribed angle is one whose vertex lies on the circle and whose sides contain chords of the circle.

Intercepted Arc Definition

An intercepted arc is the segment of the circle that lies between two lines, rays, or segments.

Theorems about Chords and Arcs

In congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicularity of Diameter and Chord

If a diameter is perpendicular to a chord, it will bisect both the chord and the arc associated with that chord.

Angle formed by Intersecting Lines

The angles formed when two lines intersect at a point on a circle relate to the arcs they intercept; the angle measure is half of that arc.

Solving for Variables in Tangents

When lines or segments that appear to be tangent are shown, they can be assumed to be tangent for solving related variables or measures.