What is minor arc?

An arc whose measure is less than 180 degrees is called a minor arc.

sparknotes.com - Geometry: Circles: Characteristics of Circles | SparkNotes

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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

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Which line is called as chord?

mathworld.wolfram.com - Chord -- from Wolfram MathWorld

In plane geometry, a chord is the line segment joining two points on a curve. The term is often used to describe a line segment whose ends lie on a circle. The term is also used in graph theory, where a cycle chord of a graph cycle is an edge not in whose endpoints lie in . In the above figure, is the radius of the circle, is the chord length, is called the apothem, and the sagitta. The shaded region in the left figure is called a circular sector, and the shaded region in the right figure is called a circular segment. There are a number of interesting theorems about chords of circles. All angles inscribed in a circle and subtended by the same chord are equal. The converse is also true: The locus of all points from which a given segment subtends equal angles is a circle.

In the left figure above,

(1)

(Jurgensen 1963, p. 345). In the right figure above,

(2)

which is a statement of the fact that the circle power is independent of the choice of the line (Coxeter 1969, p. 81; Jurgensen 1963, p. 346). Given any closed convex curve, it is possible to find a point through which three chords, inclined to one another at angles of , pass such that is the midpoint of all three (Wells 1991). Let a circle of radius have a chord at distance . The area enclosed by the chord, shown as the shaded region in the above figure, is then

(3)

But

(4)

so

(5)

and

(6) (7)

Checking the limits, when , and when ,

(8)

the expected area of the semicircle.

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