9 1 Eploring ircles A circle is the set of all points in a plane that are a given distance from a given point in the plane called the center. Various parts of a circle are labeled in the figure at the right. The distance around a circle is called the circumference. Pages 446 451 chord diameter radius ircumference of a ircle If a circle has a circumference of units and a radius of r units, then r. Eample: Find the circumference of the circle shown at the right. r (13) 6 1.7 The circumference is about 1.7 cm. 13 cm efer to S for Eercises 1 6. Y 1. Name the center of S. X. Name three radii of S. S T 3. Name a diameter. 4. Name a chord. 5. If T., find S. 6. Is S S? Eplain. In Eercises 7 10, the radius, diameter, or circumference of a circle is given. Find the other measures to the nearest tenth. 7. r 7, d?,?. d 3.4, r?, 9. 116.5, d?, r? 10. r 1, d?,?? Glencoe/cGraw-Hill 49 Geometr
9 Pages 45 45 Angles and Arcs An angle whose verte is at the center of a circle is called a central angle. A central angle separates a circle into two arcs called a major arc and a minor arc. In the circle at the right, EF is a central angle. Points and F and all points of the circle interior to EF form a minor arc called arc F. This is written F. Points and F and all points of the circle eterior to EF form a major arc called GF. You can use central angles to find both the degree measure and the length of an arc. The arcs determined b a diameter are called semicircles and have measures of 10. G major arc E F minor arc Eamples: In, m AB 4, B 1, and A is a diameter. 1 Find mab and mab. Since AB is a central angle and m AB 4, then mab 4. mab 360 mab 360 4 or 31 Find the length of AB. First, find what part of the circle is represented b AB. A B 4 7 3 60 6 0 So, the length of AB 7 is of the 6 0 circumference of. length of AB 7 (r) 6 0 6 7 0 ()(1) or about. units efer to P for Eercises 1. If SN and T are diameters with m SPT 51 and m NP 9, determine whether each arc is a minor arc, a major arc, or a semicircle. Then find the degree measure of each arc. 1. mn. mst 3. mts 4. mst If T 15, find the length of each arc. ound to the nearest tenth. 5. N 6. ST 7. TS. ST N P S T Glencoe/cGraw-Hill 50 Geometr
9 3 Arcs and hords The following theorems state relationships between arcs, chords, and diameters. In a circle or in congruent circles, two minor arcs are congruent if and onl if their corresponding chords are congruent. In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc. In a circle or in congruent circles, two chords are congruent if and onl if the are equidistant from the center. Pages 459 465 Eample: In the circle, is the center, D 15, and D 4. Find. ED 1 D 1 (4) 1 (E) (ED) (D) 1 15 144 5 1 9 E D In each circle, is the center. Find each measure. 1. mnp. K 3. XY N F H G 10 X K 1 P E Q 1 0 B Y 4. Suppose a chord is 0 inches 5. Suppose a chord of a circle long and is 4 inches from the is 5 inches from the center center of the circle. Find the and is 4 inches long. length of the radius. Find the length of the radius. 6. Suppose the diameter of a circle is 30 centimeters long and a chord is 4 centimeters long. Find the distance between the chord and the center of the circle. Glencoe/cGraw-Hill 51 Geometr
9 4 Inscribed Angles An inscribed angle of a circle is an angle whose verte is on the circle and whose sides contain chords of the circle. We sa that DEF intercepts DF. The following theorems involve inscribed angles. If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. If two inscribed angles of a circle or congruent circles intercept congruent arcs or the same arc, then the angles are congruent. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementar. E Pages 466 473 D F Eample: In the circle above, find m DEF if mdf =. Since DEF is an inscribed angle, m DEF 1 mdf 1 () or 14. In P, S TV. 1. Name the intercepted arc for TS. S. Name an inscribed angle. 3. Name a central angle. T P Q V In P, msv 6 and mps 110. Find each measure. 4. m PS 5. mt 6. m VT 7. m SVT. m TQV 9. m QT 10. m QT 11. ms Glencoe/cGraw-Hill 5 Geometr
9 5 Tangents emember that a tangent is a line in the plane of a circle that intersects the circle in eactl one point. Three important theorems involving tangents are the following. If a line is a tangent to a circle, then it is perpendicular to the radius drawn to the point of tangenc. In a plane, if a line is perpendicular to a radius of a circle at the endpoint on the circle, then the line is a tangent of the circle. If two segments from the same eterior point are tangent to a circle, then the are congruent. Pages 475 4 Eample: Find the value of if AB is tangent to. Tangent AB is perpendicular to radius B. Also, A AD B 17. (AB) (B) (A) 17 64 9 5 15 A 9 D B For each, find the value of. Assume that segments that appear to be tangent are tangent. 1.. 0 3. 19 15 14 4. 5. 6. 30 40 40 1 Glencoe/cGraw-Hill 53 Geometr
9 6 Secants, Tangents, and Angle easures A line that intersects a circle in eactl two points is called a secant of the circle. You can find the measures of angles formed b secants and tangents b using the following theorems. Pages 43 490 If a secant and a tangent intersect at the point of tangenc, then the measure of each angle formed is one-half the measure of its intercepted arc. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted b the angle and its vertical angle. If two secants, a secant and a tangent, or two tangents intersect in the eterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. Eample: Find the measure of PN. You can use the last theorem above. m(n S ) m PN 1 34 1 S N P 1 (34 1) 1 (16) or Find the measure of each numbered angle. 1.. 3. 110 5 1 40 134 3 100 0 Given T, find the value of. 4. 130 5. 6. 0 100 T T 50 T 70 Glencoe/cGraw-Hill 54 Geometr
9 7 Pages 491 497 Special Segments in a ircle The following theorems can be used to find the measure of special segments in a circle. If two chords intersect in a circle, then the products of the measures of the segments of the chords are equal. If two segments are drawn to a circle from an eterior point, then the product of the measures of one secant segment and its eternal secant segment is equal to the product of the measures of the other secant segment and its eternal secant segment. If a tangent segment and a secant segment are drawn to a circle from an eterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its eternal secant segment. Eample: Find the value of to the nearest tenth. (AB) B BD Theorem 9 16 (1) 15 (15 ) 34 5 15 99 15 6.6 Find the value of to the nearest tenth. Assume segments that appear tangent to be tangent. D A 1 15 B 1.. 3. 3 6 6 7.3 5. 4. 5. 6. 3 6 6 7 4 14 16 7.. 9. 3 13 9 4 6 5 3 4 Glencoe/cGraw-Hill 55 Geometr
9 Integration: Algebra Equations of ircles The standard equation for a circle is derived from using the distance formula given the coordinates of the center of the circle and the measure of its radius. An equation for a circle with center (h, k) and a radius of r units is ( h) ( k) r. Eample: Graph the circle whose equation is ( 3) ( 1) 16. Pages 49 503 ( h) ( k) r standard equation ( (3)) ( 1) (16) rewrite equation in standard form 4 (3, 1) Therefore, h 3, k 1, and r 16 4. The center is at ( 3, 1) and the radius is 4 units. Determine the coordinates of the center and the measure of the radius for each circle whose equation is given. 1. ( 7.) ( 3.4) 14.44. 1 ( ) 1 6 5 3. ( 6) ( 3) 5 0 Graph each circle whose equation is given. Label the center and measure of the radius on each graph. 4. (.5) ( 1) 1.5 5. ( 3) ( 4).5 0 6. 1 3 4 1 7. ( ) 9 Glencoe/cGraw-Hill 56 Geometr