Lesson 5 Circles

Definitions A circle is the set of all points in a plane that are the same distance from a fixed point called the center of the circle. A radius of a circle is a line segment extending from the center to the circle. A diameter is a line segment that joins two points on the circle and passes through the center. radius center diameter

Naming a Circle A circle in a diagram is named by its center. The circle at right is called circle O or: If there is more than one circle in a diagram with the same center, this notation does not suffice. Note: two circles in the same plane with the same center are called concentric circles. O

The word radius (plural: radii) is also used to denote the length of a radius (all radii have the same length). The word diameter is also used to denote the length of a diameter (all diameters have the same length). Note that the diameter of a circle is twice its radius.

Chords A chord is any line segment that joins two points on a circle. Therefore, a diameter is an example of a chord. It is the longest possible chord.

Chords and Radii Given a chord in a circle, any radius that bisects the chord (passes through its midpoint) is perpendicular to that chord. Also, if a radius is perpendicular to a chord, then it bisects the chord.

Tangents Given a circle, a line is tangent to the circle if it touches it just once. Such a line is called a tangent or a tangent line. The point where the tangent touches the circle is called the point of tangency. We can also speak of tangent segments or rays. One crucial property of tangents is that the radius drawn to the point of tangency is perpendicular to the tangent.

Secants Given a circle, a line that intersects the circle twice is called a secant line or a secant. Line segments and rays may also be secants. Often, secants are drawn from a point outside of the circle to a point on the circle.

Arcs An arc is an unbroken part of a circle. For example, in the figure, the part of the circle shaded red is an arc. A semicircle is an arc equal to half a circle. A minor arc is smaller than a semicircle. A major arc is larger than a semicircle.

Naming Arcs A minor arc, like the one in red in the figure, can be named by drawing an arc symbol over its endpoints: Sometimes, to avoid confusion, a third point between the endpoints is used to name the arc: The reason for this is to avoid ambiguity because, given two points on a circle, there are two arcs between them (the long way around the circle or the short way). A P B

The arc highlighted in red in the figure would be called Semicircles and major arcs must be named with three (or sometimes more) points. The arc highlighted in red in the figure would be called It appears to be a major arc. If we wrote then we would be referring to the part of the circle that is not highlighted in red (a minor arc, it seems). A B C

The Measure of an Arc Each arc has a degree measure between 0 degrees and 360 degrees. A full circle is 360 degrees, a semicircle is 180 degrees, a minor arc measures less than 180 degrees, and a major arc measures more than 180 degrees. If an arc is a certain fraction of a circle, then its measure is the same fraction of 360 degrees. Some sample arc measures are given below. C A E B D F G H

Example In the figure, and Find Let denote the measure of each of the two equal arcs. Then P Q R

Central Angles Given a circle, a central angle is an angle whose vertex is at the center of the circle. In the figure, the center of the circle is and is a central angle that intercepts arc The measure of a central angle is equal to the measure of the arc it intercepts. P A Q B

Inscribed Angles In a circle, an inscribed angle is an angle whose vertex is on the circle and whose sides are chords. In the figure, is an inscribed angle and it intercepts arc The measure of an inscribed angle is half the measure of its intercepted arc. P A B Q