10-1 Study Guide and Intervention

Transcription

1 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-1 tudy Guide and Intervention ircles and ircumference arts of ircles circle consists of all points in a plane that are a given distance, called the radius, from a given point called the center. segment or line can intersect a circle in several ways. segment with endpoints that are the center of the circle and a point of the circle is a radius. segment with endpoints that lie on the circle is a chord. chord that contains the circle s center is a diameter. F chord:, radius: F, F, F diameter: ample a. Name the circle. he name of the circle is. b. Name radii of the circle.,,, and are radii. c. Name chords of the circle. and are chords. d. Name a diameter of the circle. is a diameter. ercises 1. Name the circle. X. Name radii of the circle. 3. Name chords of the circle. Y 4. Name diameters of the circle. 5. Find if is 1 millimeters.. Find and if Y is 10 inches. 7. Is XY? plain. hapter 10 Glencoe Geometry

2 NM I 10-1 tudy Guide and Intervention (continued) ircles and ircumference ircumference he circumference of a circle is the distance around the circle. ircumference For a circumference of units and a diameter of d units or a radius of r units, d or r. ample Find the circumference of the circle to the nearest hundredth. r ircumference formula (13) r Use a calculator. he circumference is about 1. centimeters. ercises 13 cm Lesson 10-1 Find the circumference of a circle with the given radius or diameter. ound to the nearest hundredth. 1. r cm. r 3 ft 3. r 4.1 cm 4. d 10 in. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 5. d 1 m. d 1 yd 3 he radius, diameter, or circumference of a circle is given. Find the missing measures to the nearest hundredth. 7. r 4 cm. d ft d, r, 9. r 1 cm 10. d 15 in. d, r, Find the eact circumference of each circle cm cm 1 cm cm hapter 10 7 Glencoe Geometry

3 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 10- tudy Guide and Intervention ample In, m 4 and is a diameter. Find m and m. is a central angle and m 4, so m 4. hus m 30 4 or 31. ercises NM I Measuring ngles and rcs ngles and rcs central angle is an angle whose verte is at the center of a circle and whose sides are radii. central angle separates a circle into two arcs, a major arc and a minor arc. G Here are some properties of central angles and arcs. he sum of the measures of the central angles of mh mf mfg mgh 30 a circle with no interior points in common is 30. he measure of a minor arc equals the measure of its central angle. he measure of a major arc is 30 minus the measure of the minor arc. wo arcs are congruent if and only if their corresponding central angles are congruent. he measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. (rc ddition ostulate) Find each measure. 1. m. mu 3. m 4. m In, m 44. Find each measure. 5. m. m 7. m. m 9. m 10. m H mf mf mgf 30 mf F FG if and only if F FG. mf mfg mg 45 0 F GF is a minor arc. HG is a major arc. U GF is a central angle. Lesson 10- hapter Glencoe Geometry

4 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10- tudy Guide and Intervention (continued) Measuring ngles and rcs rc Length n arc is part of a circle and its length is a part of the circumference of the circle. ample In, m 135,, and is a diameter. Find the length of. m 135, so m 135. Using the formula r, the circumference is () or 1. o find the length of, write a proportion to compare each part to its whole. length of degree measure of arc circumference degree measure of circle roportion ubstitution (1 )( 135) 30 Multiply each side by 1. implify. he length of is or about 1.5 units. ercises he diameter of is 4 units long. Find the length of each arc for the given angle measure. ound to the nearest tenth. 1. if m 10. if m if m if m 45 he diameter of is 15 units long and. Find the length of each arc for the given angle measure. ound to the nearest tenth. 5. if m 70 M. N if m 50 N 7. M. M if mm 140 hapter Glencoe Geometry

5 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-3 tudy Guide and Intervention (continued) rcs and hords rcs and hords oints on a circle determine both chords and arcs. everal properties are related to points on a circle. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. If all the vertices of a polygon lie on a circle, the polygon is said to be inscribed in the circle and the circle is circumscribed about the polygon. V V if and only if V. V is inscribed in. is circumscribed about V. ample rapezoid is inscribed in. If and m 50, what is m? hords,, and are congruent, so,, and are congruent. m 50, so m m m hen m or 10. ercises ach regular polygon is inscribed in a circle. etermine the measure of each arc that corresponds to a side of the polygon. 1. heagon. pentagon 3. triangle 4. square 5. octagon. 3-gon etermine the measure of each arc of the circle circumscribed about the polygon U 9. V U 4 V 7 U hapter 10 0 Glencoe Geometry

6 NM I 10-3 tudy Guide and Intervention (continued) rcs and hords iameters and hords 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 only if they are equidistant from the center. W Z X Y If WZ, then X X and W W. If X Y, then. If, then and are equidistant from point. ample In,, 15, and 4. Find. diameter or radius perpendicular to a chord bisects the chord, so is half of. 1 (4) 1 Use the ythagorean heorem to find in. () () () ythagorean heorem ubstitution Multiply. ubtract 144 from each side. ake the square root of each side. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. ercises In, 4 and my 45. Find each measure my 7. mx. mx. m 9. m In G, G GU and. Find each measure. 10. U m G 15. m 5 G 3 U X Y Lesson chord of a circle 0 inches long is 4 inches from the center of a circle. Find the length of the radius. hapter 10 1 Glencoe Geometry

7 NM I 10-4 tudy Guide and Intervention Inscribed ngles Inscribed ngles n inscribed angle is an angle whose verte is on a circle and whose sides contain chords of the circle. In G, inscribed F intercepts F. G Inscribed ngle heorem If an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted arc. F mf 1 mf ample In G above, mf 90. Find mf. F is an inscribed angle so its measure is half of the intercepted arc. mf 1 mf 1 (90) or 45 ercises Use for ercises In, V and V. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 1. Name the intercepted arc for.. Name an inscribed angle that intercepts V. In, mv 10 and m 7. Find each measure. 3. m 4. mv 5. m. mv 7. m. mv 9. mv 10. mv V Lesson 10-4 hapter 10 7 Glencoe Geometry

8 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-4 tudy Guide and Intervention (continued) Inscribed ngles ngles of Inscribed olygons n inscribed polygon is one whose sides are chords of a circle and whose vertices are points on the circle. Inscribed polygons have several properties. If an angle of an inscribed polygon intercepts a semicircle, the angle is a right angle. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. If is a semicircle, then m 90. For inscribed quadrilateral, m m 10 and m m 10. ample a. m In above, 3 and 5. Find each measure. intercepts a semicircle. herefore is a right angle and m 90. b. is a right triangle, so use the ythagorean heorem to find. () () () () 3 5 () 5 9 () 1 4 ercises Find the measure of each angle or segment for each figure. 1. mx, my. 3. m1, m X G W 10 Y 5 F Z 1 J H 4. m1, m 5.,. m1, m K U M N 30 Z L 5 30 W 1 V hapter 10 Glencoe Geometry

9 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-5 tudy Guide and Intervention (continued) angents angents tangent to a circle intersects the circle in eactly one point, called the point of tangency. here are three important relationships involving tangents. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is a tangent to the circle. If two segments from the same eterior point are tangent to a circle, then they are congruent. if and only if is tangent to. If and are tangent to, then. ample is tangent to. Find. is tangent to, so is perpendicular to radius. is a radius, so and 9 or 17. Use the ythagorean heorem with right. 9 () () () ythagorean heorem ubstitution Multiply. ubtract 4 from each side. ake the positive square root of each side. ercises Find. ssume that segments that appear to be tangent are tangent. 1.. J 0 K H G F 3. N 4. 1 M U Y Z 5 F hapter Glencoe Geometry

10 NM I 10-5 tudy Guide and Intervention (continued) angents ircumscribed olygons When a polygon is circumscribed about a circle, all of the sides of the polygon are tangent to the circle. G H F Heagon F is circumscribed about.,,,, F, and F are tangent to. K J quare GHJK is circumscribed about. GH, JH, JK, and KG are tangent to. ample is circumscribed about. Find the perimeter of. is circumscribed about, so points,, and F are points of tangency. herefore F,, and F. F F F he perimeter is 5 units. opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. ercises Find. ssume that segments that appear to be tangent are tangent. 1.. square square regular heagon 4 4 equilateral triangle Lesson 10-5 hapter Glencoe Geometry

11 NM I 10- tudy Guide and Intervention ecants, angents, and ngle Measures Intersections n or Inside a ircle line that intersects a circle in eactly two points is called a secant. he measures of angles formed by secants and tangents are related to intercepted arcs. If two secants intersect in the interior of a circle, then the measure of the angle formed is one-half the sum of the measure of the arcs intercepted by the angle and its vertical angle. 1 m1 1 (m m ) If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc. X Y U V mxv 1 muv myv 1 mv Lesson 10- ample 1 Find. ample Find y. 30 he two secants intersect he secant and the inside the circle, so is tangent intersect at the equal to one-half the sum point of tangency, so the of the measures of the arcs 55 measure of the angle is intercepted by the angle one-half the measure of and its vertical angle. its intercepted arc. 1 y 1 (30 55) y 1 (1) opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 1 (5) 4.5 ercises Find each measure m1. m 3. m m4 5. m5. m 3 0 U 10 4 V W X hapter Glencoe Geometry

12 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10- tudy Guide and Intervention (continued) ecants, angents, and ngle Measures Intersections utside a ircle If secants and tangents intersect outside a circle, they form an angle whose measure is related to the intercepted arcs. 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. H G J K N M and are secants. G is a tangent. J is a secant. mgh m 1 (m m ) m 1 (mgkj ) m 1 M and N are tangents. (mmn mmn ) ample Find mmn. MN is formed by two secants that intersect in the eterior of a circle. mmn 1 (mmn m) M 34 N 1 1 (34 1) 1 (1) or he measure of the angle is. ercises Find each measure. 1. m1. m m hapter Glencoe Geometry

13 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10-7 tudy Guide and Intervention pecial egments in a ircle egments Intersecting Inside a ircle If two chords intersect in a circle, then the products of the measures of the chords are equal. a c d b a b c d ample Find. he two chords intersect inside the circle, so the products and are equal. 3 ubstitution 4 implify. 4 ivide each side by. 3 ercises Find to the nearest tenth hapter Glencoe Geometry

14 NM I 10-7 tudy Guide and Intervention (continued) pecial egments in a ircle egments Intersecting utside a ircle If secants and tangents intersect outside a circle, then two products are equal. If two secant 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. and are secant segments. and are eternal secant segments. 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. is a tangent segment. is a secant segment. is an eternal secant segment. () Lesson 10-7 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. ample Find to the nearest tenth. he tangent segment is, the secant segment is, and the eternal secant segment is. () (1) 15(15 ) ercises Find to the nearest tenth. ssume segments that appear to be tangent are tangent W Y 3 V hapter Glencoe Geometry

15 NM I 10- tudy Guide and Intervention quations of ircles quation of a ircle circle is the locus of points in a plane equidistant from a given point. You can use this definition to write an equation of a circle. tandard quation n equation for a circle with center at (h, k) of a ircle and a radius of r units is ( h) (y k) r. y r (h, k) ample Write an equation for a circle with center (1, 3) and radius. Use the formula ( h) ( y k) r with h 1, k 3, and r. ( h) ( y k) r quation of a circle ( (1)) ( y 3) ubstitution ( 1) ( y 3) 3 implify. ercises Write an equation for each circle. 1. center at (0, 0), r. center at (, 3), r 5 Lesson 10- opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. 3. center at (, 4), r 1 4. center at (1, 4), r 5. center at (, ), diameter. center at 1, 1 4, r 3 7. center at the origin, diameter 4. center at 1, 5, r 5 9. Find the center and radius of a circle with equation y Find the center and radius of a circle with equation ( 4) (y 3) 1. hapter Glencoe Geometry

16 opyright Glencoe/McGraw-Hill, a division of he McGraw-Hill ompanies, Inc. NM I 10- tudy Guide and Intervention (continued) quations of ircles Graph ircles If you are given an equation of a circle, you can find information to help you graph the circle. ample Graph ( 3) (y 1) 9. Use the parts of the equation to find (h, k) and r. y ( h) ( y k) r ( h) ( 3) ( y k) ( y 1) r 9 h 3 y k y 1 r 3 h 3 k 1 h 3 k 1 (3, 1) he center is at (3, 1) and the radius is 3. Graph the center. Use a compass set at a radius of 3 grid squares to draw the circle. ercises Graph each equation. 1. y 1. ( ) ( y 1) 9 y y 3. ( ) y 1 4. ( 1) ( y ).5 y y 5. 1 y ( y 1) 9 y y hapter 10 5 Glencoe Geometry