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Geometry: Circles

Characteristics of Circles

Problems

Problems

Special names are given to geometric figures that lie on or inside circles. Among these geometric figures are arcs, chords, sectors, and segments.

Arc

The arc of a circle consists of two points on the circle and all of the points on the circle that lie between those two points. It's like a segment that was wrapped partway around a circle. An arc is measured not by its length (although it can be, of course) but most often by the measure of the angle whose vertex is the center of the circle and whose rays intercept the endpoints of the arc. Hence an arc can be anywhere from 0 to 360 degrees. Below an arc is pictured.

Figure %: An arc

The arc above contains points A, B, and all the points between them. But what if the arc went the other way around the circle? This brings up an important point. Every pair of endpoints defines two arcs. An arc whose measure is less than 180 degrees is called a minor arc. An arc whose measure is greater than 180 degrees is called a major arc. An arc whose measure equals 180 degrees is called a semicircle, since it divides the circle in two. Every pair of endpoints on a circle either defines one minor arc and one major arc, or two semicircles. Only when the endpoints are endpoints of a diameter is the circle divided into semicircles. From this point on, unless otherwise mentioned, when arcs are discussed you may assume the arc is a minor arc.

Figure %: A major arc, minor arc, and semicircle

A central angle is an angle whose vertex is the center of a circle. Any central angle intercepts the circle at two points, thus defining an arc. The measure of a central angle and the arc it defines are congruent.

Figure %: A central angle and the arc it defines

Chord

A chord is a segment whose endpoints are on a circle. Thus, a diameter is a special chord that includes the center.

Figure %: A chord

Chords have a number of interesting properties. Every chord defines an arc whose endpoints are the same as those of the chord. For example, a diameter and semicircle are a chord and arc that share the same endpoints. The union of a chord with a central angle forms a triangle whose sides are the chord and the two radii that lie in the rays that make up the angle. This kind of triangle is always an isosceles triangle--we'll define that term in Geometry 2. Also, the diameter perpendicular to a given chord (remember, there is only one such diameter because a diameter must contain the center) is also the perpendicular bisector of that chord. These ideas are illustrated below.

Figure %: Properties of chords

Sectors and Segments

Central angles and chords also define certain regions within a circle. These regions are called sectors and segments. A sector of a circle is the region enclosed by the central angle of a circle and the circle itself. A segment of a circle is the region enclosed by a chord and the arc that the chord defines. A given segment is always a subregion of the sector defined by the central angle that intersects the circle at the endpoints of the chord that defines the given segment. Sound a little complicated? It isn't. Take a look at the drawing.

Figure %: A sector and a segment of a circle
The sector is the region shaded in on the left. The rays of the central angle DCE and the arc DE enclose the sector. The segment of the circle, which is shaded in on the right side of the circle, is bounded by the chord AB and the arc AB. Were the central angle ACB to be drawn, a sector would be defined that would include all of the segment created by the chord AB.

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