Study Guide and Intervention - prosseracademy.org

1 ExercisesExampleChapter 106 Glencoe Geometry10-1 Study Guide and InterventionCircles and CircumferenceNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, of CirclesA circleconsists of all points in a plane that are a given distance, called the radius, from a given point called the segment or line can intersect a circle in several ways. A segment with endpoints that are the center of the circle and a point of the circle is a radius. A segment with endpoints that lie on the circle is a chord. A chord that contains the circle s center is a Name the name of the circle is Name radii of the O ,B O ,C O , and D O are Name chords of the B and C D are Name a diameter of the B is a the radii of the chords of the diameters of the ARif ABis 18 ARand ABif RYis 10 A B X Y ?

2 : A E , B D radius: F B , F C , F D diameter: B D ABCDEF1-62 Geo-10-873967 5/16/06 2:23 PM Page 6 ExercisesExampleChapter 107 Glencoe GeometryLesson 10-110-1 Study Guide and Intervention (continued)Circles and CircumferenceNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, circumferenceof a circle is the distance around the a circumference of Cunits and a diameter of dunits or a radius of runits, C dor C 2 the circumference of the circle to the nearest 2 rCircumference formula 2 (13)r 13 a circumference is about the circumference of a circle with the given radius or diameter. Round to thenearest 8 3 2 10 13 18 ydThe radius, diameter, or circumference of a circle is given.

3 Find the missingmeasures to the nearest 4 6 ftd ,C r ,C 12 15 ,C r ,C Find the exact circumference of each cm 2 cm 12 cm5 cm13 cm1-62 Geo-10-873967 5/16/06 10:04 AM Page 7 ExercisesExampleLesson 10-2 Chapter 1013 Glencoe Geometry10-2 Study Guide and InterventionMeasuring Angles and ArcsNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, and ArcsA central angleis an angle whose vertex is at the center of a circle and whose sides are radii. A central angle separates a circle into two arcs, a major arcand a minor are some properties of central angles and arcs. The sum of the measures of the central angles of m HEC m CEF m FEG m GEH 360a circle with no interior points in common is 360.

4 The measure of a minor arc equals the measure mCF m CEFof its central angle. The measure of a major arc is 360 minus the mCGF 360 mCF measure of the minor arc. Two arcs are congruent if and only if their CF FG if and only if CEF central angles are congruent. The measure of an arc formed by two adjacent mCF mFG mCG arcs is the sum of the measures of the two arcs.(Arc Addition Postulate)In R,m ARB 42 and A C is a mAB and mACB . ARBis a central angle and m ARB 42, so mAB mACB 360 42 or each QCTIn O,m BOA 44. Find each ADCBOTUQRS60 45 CBCARGF is a minor is a major arc. GEFis a central Geo-10-873967 4/19/06 10:26 AM Page 13 Chapter 1014 Glencoe Geometry10-2 Study Guide and Intervention (continued)Measuring Angles and ArcsNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, LengthAn arc is part of a circle and its length is a part of the circumference of the R,m ARB 135,RB 8, and A C is a diameter.

5 Find the length of AB .m ARB 135, so mAB 135. Using the formula C 2 r, the circumference is 2 (8) or 16 . To find the length of AB , write a proportion to compare each part to its whole. Proportion 16 133650 Substitution (16 36)(0135) Multiply each side by 16 . 6 length of AB is 6 or about diameter of Ois 24 units long. Find the length of each arc for the given angle measure. Round to the nearest if m DOE if m DOE if m COB if m COB 45 The diameter of Pis 15 units long and SPT the length of each arc for the given angle to the nearest if m SPT if m RPT if m MPS 140 RNPSMTACDBEO degree measure of arc degree measure of circlelength of AB circumferenceACBRE xampleExercises1-62 Geo-10-873967 4/19/06 10:27 AM Page 14 ExercisesExampleArcs and ChordsPoints on a circle determine both chords and arcs.

6 Several 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 TV if and only if R S T V . If all the vertices of a polygon lie on a circle, the polygon RSVTis inscribed in said to be inscribedin the circle and the circle is Ois circumscribed about the ABCDis inscribed in A B B C C D and mBC 50, what is mAPD ?Chords A B ,B C , and C D are congruent, so AB ,BC , and CD are 50, so mAB mBC mCD 50 50 50 150. Then mAPD 360 150 or regular polygon is inscribed in a circle. Determine the measure of each arcthat corresponds to a side of the the measure of each arc of the circle circumscribed about the 1020 Glencoe Geometry10-3 Study Guide and Intervention (continued)Arcs and ChordsNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Geo-10-873967 4/19/06 10:30 AM Page 20 ExercisesExampleDiameters and Chords In a circle, if a diameter is perpendicular to a chord, then it bisects the chord and its arc.

7 In a circle or in congruent circles, two chords are congruent if and only if they areequidistant from the W Z A B , then A X X B and AW WB .If OX OY, then A B R S .If A B R S , then A B and R S are equidistant from point O,C D O E ,OD 15, and CD 24. Find diameter or radius perpendicular to a chord bisects the chord,so EDis half of 12 (24) 12 Use the Pythagorean Theorem to find xin OED.(OE)2 (ED)2 (OD)2 Pythagorean Theoremx2 122 152 Substitutionx2 144 81 Subtract 144 from each 9 Take the square root of each P,CD 24 and mCY 45. Find each In G,DG GUand AC RT. Find each chord of a circle 20 inches long is 24 inches from the center of a circle.

8 Find the length of the 10-3 Chapter 1021 Glencoe Geometry10-3 Study Guide and Intervention (continued)Arcs and ChordsNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Geo-10-873967 4/19/06 10:31 AM Page 21 ExampleExercisesLesson 10-4 Chapter 1027 Glencoe Geometry10-4 Study Guide and InterventionInscribed AnglesNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, AnglesAn inscribed angleis an angle whose vertex is on a circle and whose sides contain chords of the circle. In G,inscribed DEFinterceptsDF .Inscribed Angle TheoremIf an angle is inscribed in a circle, then the measure of the angle equals one-half the measure of its intercepted DEF 12 mDF In Gabove,mDF 90.

9 Find m DEF. DEFis an inscribed angle so its measure is half of the intercepted DEF 12 mDF 12 (90) or 45 Use Pfor Exercises 1 10. In P,R S ||T V and R T S V . the intercepted arc for an inscribed angle that intercepts SV .In P,mSV 120 and m RPS 76. Find each SVTPQRSTVDEFG1-62 Geo-10-873967 4/19/06 10:34 AM Page 27 ExercisesExampleChapter 1028 Glencoe Geometry10-4 Study Guide and Intervention (continued)Inscribed AnglesNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, of Inscribed PolygonsAn inscribed polygonis one whose sides are chords of a circle and whose vertices are points on the circle.

10 Inscribed polygonshave several properties. If an angle of an inscribed polygon intercepts a If BCD is a semicircle, then m BCD , the angle is a right angle. If a quadrilateral is inscribed in a circle, then its For inscribed quadrilateral ABCD,opposite angles are A m C 180 andm ABC m ADC Rabove,BC 3 and BD 5. Find each C Cintercepts a semicircle. Therefore Cis a right angle and m C BCDis a right triangle, so use thePythagorean Theorem to find CD.(CD)2 (BC)2 (BD)2(CD)2 32 52(CD)2 25 9(CD)2 16CD 4 Find the measure of each angle or segment for each X,m 1,m 1,m , 1,m 292 21 ZWTUV30 30 33 SRDABC2165 PQMKNL2140 EFGHJ12 DABC5 ZWXY120 55 1-62 Geo-10-873967 4/19/06 10:35 AM Page 28 ExercisesExampleChapter 1034 Glencoe Geometry10-5 Study Guide and Intervention (continued)TangentsNAME _____ DATE _____ PERIOD _____Copyright Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, tangent to a circle intersects the circle in exactly one point, called the point of tangency.