Circles and Related Segments and Angles 1

Circles and Related Segments and Angles 6.1 . Circles and Related Segments and Angles Definition A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. 3 Circles and Related Segments and Angles A circle is named by its center point. In Figure 6.1, point P is the center of the circle. The symbol for circle is , so the circle in Figure 6.1 is P. Points A, B, C, and D are points of (or on) the circle. Points P (the center) and R are in the interior of circle P; points G and H are in the exterior of the circle. Figure 6.1 4 Circles and Related Segments and Angles In Q of Figure 6.2, is a radius of the circle. A radius is a segment that joins the center of the circle to a point on the circle. , and are radii (plural of radius) of definition, SQ = TQ = VQ = WQ. Figure 6.2 Q. By 5 Circles and Related Segments and Angles The following statement is a consequence of the definition of a circle. A line segment (such as in Figure 6.2) that joins two points of a circle is a chord of the circle. A diameter of a circle is a chord that contains the center of the circle; if T-Q-W in Figure 6.2, then is a diameter of Q. 6 Circles and Related Segments and Angles Definition Congruent circles are two or more circles that have congruent radii. In Figure 6.3, circles P and Q are congruent because their radii have equal lengths. We can slide P to the right to coincide with Q. (a) (b) Figure 6.3 7 Circles and Related Segments and Angles Definition Concentric circles are coplanar circles that have a common center. The concentric circles in Figure 6.4 have the common center O. Figure 6.4 8 Circles and Related Segments and Angles In P of Figure 6.5, the part of the circle shown in red from point A to point B is arc AB, symbolized by . If is a diameter, then (three letters are used for clarity) is a semicircle. In Figure 6.5, a minor arc like is part of a semicircle; a major arc such as (also denoted by or ) is more than a semicircle but less than the entire circle. Figure 6.5 9 Circles and Related Segments and Angles Definition A central angle of a circle is an angle whose vertex is the center of the circle and whose sides are radii of the circle. In Figure 6.6, NOP is a central angle of intercepted arc of NOP is O. The Figure 6.6 10 Circles and Related Segments and Angles The intercepted arc of an angle is determined by the two points of intersection of the angle with the circle and all points of the arc in the interior of the angle. 11 Example 1 In Figure 6.6, and intersect at O, the center of the circle. Name: a) All four radii b) Both diameters c) All four chords d) One central angle e) One minor arc f) One semicircle g) One major arc Figure 6.6 h) Intercepted arc of MON i) Central angle that intercepts 12