Copyright © Cengage Learning. All rights reserved. 12 Geometry

Copyright © Cengage Learning. All rights reserved. Circles 12.5

3 Circles A circle is a plane curve consisting of all points at a given distance (called the radius, r) from a fixed point in the plane, called the center. (See Figure ) The diameter, d, of the circle is a line segment through the center of the circle with endpoints on the circle. Note that the length of the diameter equals the length of two radii—that is, d = 2r. Figure Circle

4 Circles The circumference of a circle is the distance around the circle. The ratio of the circumference of a circle to the length of its diameter is a constant called  (pi). The number  cannot be written exactly as a decimal. Decimal approximations for  are 3.14 or When solving problems with , use the  key on your calculator.

5 Circles The following formulas are used to find the circumference and the area of a circle. C is the circumference and A is the area of a circle; d is the length of the diameter, and r is the length of the radius.

6 Example 1 Find the area and the circumference of the circle shown in Figure The formula for the area of a circle given the radius is A =  r 2 Figure 12.45

7 Example 1 A =  (16.0 cm) 2 A = 804 cm 2 The formula for the circumference of a circle given the radius is C = 2  r C = 2  (16.0 cm) = 101 cm cont’d

8 Circles An angle whose vertex is at the center of a circle is called a central angle. Angle A in Figure is a central angle. In general, Central angle Figure 12.46

9 Common Terms and Relationships of a Circle

10 Common Terms and Relationships of a Circle A chord is a line segment that has its endpoints on the circle. A secant is any line that intersects a circle at two points. A tangent is a line that has only one point in common with a circle and lies totally outside the circle. In Figure 12.47, C is the center. is a chord. Line n is a secant. Line m is a tangent. is a diameter. is a radius. Figure 12.47

11 Arcs

12 Arcs An inscribed angle is an angle whose vertex is on the circle and whose sides are chords. The part of the circle between the two sides of an inscribed or central angle is called the intercepted arc. In Figure 12.48, C is the center and ACB is a central angle. DEF is an inscribed angle. is the intercepted arc of ACB. is the intercepted arc of DEF. Figure Arcs of a circle

13 Arcs The following three relationships are often helpful to solve problems: The measure of a central angle in a circle is equal to the measure of its intercepted arc. (See Figure ) The measure of an inscribed angle in a circle is equal to one-half the measure of its intercepted arc. (See Figure ) Figure 12.49

14 Arcs The measure of an angle formed by two intersecting chords in a circle is equal to one-half the sum of the measures of the intercepted arcs. (See Figure ) Figure 12.50

15 Other Chords and Tangents

16 Other Chords and Tangents A diameter that is perpendicular to a chord bisects the chord. (See Figure ) Figure 12.51

17 Other Chords and Tangents A line segment from the center of a circle to the point of tangency is perpendicular to the tangent. (See Figure ) Figure 12.52

18 Other Chords and Tangents Two tangents drawn from a point outside a circle to the circle are equal. The line segment drawn from the center of the circle to this point outside the circle bisects the angle formed by the tangents. (See Figure ) Figure 12.53