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Name12-1ClassDateAdditional Vocabulary SupportTangent LinesComplete the vocabulary chart by filling in the missing information.Word orWord PhraseDefinitionPicture or ExamplecircleA circle is the set of all points thatare the same distance from thecenter point.radiusA radius is a line segment with oneendpoint at the center of a circleand the other endpoint at any pointon the circle.1.tangentA line is tangent to a circle if itintersects a circle at exactlyone point.2.intersect3. Two lines or figures intersect ifthey have one or more pointsin common.perpendicular4. Perpendicular lines are two linesthat intersect each other and formright angles.PythagoreanTheoremThe Pythagorean Theorem states thatin a right triangle the square of thelength of the hypotenuse is equal tothe sum of the squares of the lengthsof the other two sides.5.inscribedA circle is inscribed in a polygonif the sides of the polygon aretangent to the circle.6.circumscribed7. A circle is circumscribed in apolygon if the vertices of thepolygon are on the circle.cabPrentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.1a2 b2 c2

Name12-1ClassDateThink About a PlanTangent LinesBa. A belt fits snugly around the two circular pulleys.CE is an auxiliary line from E to BD. CE 6 AB.14 in.What type of quadrilateral is ABCE? Explain.C35 in.DA8 in.Eb. What is the length of CE?c. What is the distance between the centers of the pulleys to the nearest tenth?1. What is the definition of a tangent line? a line that touches a circle at only one point2. What is the relationship between a line tangent to a circle and the radius at thepoint of tangency (Theorem 12-1)? They are perpendicular.3. Where is the point of tangency for AB on (D? On (E? B; A4. What is the measure of /CBA? What is the measure of /BAE? Explain.908; 908; they are formed by perpendicular line segments.5. How can you use parallel lines to find the measure of /CEA?Because CE n AB, lBAE and lCEA are supplementary angles, so mlCEA 5 908.6. How can you use parallel lines or the Polygon Angle-Sum Theorem to find themeasure of /BCE?Because CE n AB, lCBA and lBCE are supplementary angles, so mlBCE 5 908. Or,lBCE is the fourth angle in a quadrilateral in which the other angles sum to 2708,so its measure is 908.7. What type of quadrilateral has four right angles? a rectangle8. What is the length of BA? 35 in.9. What is the length of CE? 35 in.10. What are the center points of the pulleys? D; E11. How can you use the Segment Addition Postulate to find CD?BD 2 BC 5 CD, BD 5 14 in., and BC 5 8 in.12. What is the measure of CD? 6 in.13. How can you use the Pythagorean Theorem to find the length of DE?a2 1 b2 5 c2 ; if CD 5 a, CE 5 b, and DE 5 c, then DE 5 "(62) 1 (352) 5 "1261 N 35.5 in.Prentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.2

NameClass12-1DatePracticeForm GTangent LinesAlgebra Assume that lines that appear to be tangent are tangent. O is thecenter of each circle. What is the value of x?1.140O x 2.39x 40 3.20O70 Ox 51 The circle at the right represents Earth. The radius of theEarth is about 6400 km. Find the distance d that a person cansee on a clear day from each of the following heights h aboveEarth. Round your answer to the nearest tenth of a kilometer.4. 12 km 392.1 km5. 20 km 506.4 kmhdrr6. 1300 km 4281.4 kmIn each circle, what is the value of x to the nearest tenth?7.O8.3.75x9Oxx69.5820xxx10.71212Determine whether a tangent line is shown in each diagram. Explain.10.11.3 63312.12610.6104.5no; 4.52 1 102 u 122yes;321(3"3)256295.6yes; 5.62 1 92 5 10.6213. TY and ZW are diameters of (S. TU and UXare tangents of (S. What is m/SYZ? 6132 TWSZPrentice Hall Gold Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.3UXY

NameClass12-1DatePractice (continued)Form GTangent LinesEach polygon circumscribes a circle. What is the perimeter of each polygon?14.3 mm15.70 mm42 in.10 in.8 mm17 mm6 in.7 mm5 in.16.7 in.17.28 in.72 ft15 ft3 in.21 ft5 in.24 ft4 in.18. Error Analysis A classmate states that BC is tangentto (A. Explain how to show that your classmate is wrong.AIf BC is tangent to (A, then AB ' BC and mlB 5 90; thiscannot be true because the sum of the three angles wouldbe greater than 1808.67 B19. The peak of Mt. Everest is about 8850 m above sea level. About how6400640016 in.20. The design of the banner at the right includes6 in.6 in.9 in.6 in.9 in.12 in.6 in.16 in.Prentice Hall Gold Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.428 8.85many kilometers is it from the peak of Mt. Everest to the horizonif the Earth’s radius is about 6400 km? Draw a diagram to help yousolve the problem. 337 kma circle with a 12-in. diameter. Using themeasurements given in the diagram,explain whether the lines shown aretangents to the circle. no; 122 1 162 u 212Cd

NameClassDatePractice12-1Form KTangent LinesLines that appear to be tangent are tangent. O is the center of each circle.What is the value of x?1.To start, identify the type of geometric figureformed by the tangent lines and radii.145x OThe figure formed is a 9. quadrilateral35 542.3.x 1653 OOx 36 The circle at the right represents Earth. The radius of Earth is about 6400 km.Find the distance d to the horizon that a person can see on a clear day fromeach of the following heights h above Earth. Round your answer to thenearest tenth of a kilometer.4. 7 km299.4 km5. 400 km2297.8 km6. 2000 m160.0 kmAlgebra In each circle, what is the value of x to the nearest tenth?7.6.812xxTo start, use the Pythagorean Theorem.x2 1 122 5 (9)2 x 1 773.9 in.8.9.7 m9.xxx16 mx5 in.9m8 in.10. QO and UR are diameters of (P .RRS and TS are tangents of (P .PFind m/UPT and m/UQP. 32; 53 QO16 SUTPrentice Hall Foundations Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.5hdrr

NameClass12-1DatePractice (continued)Form KTangent LinesDetermine whether a tangent is shown in each diagram. Explain.To start, use the Converse of the PythagoreanTheorem to relate the side lengths of the triangle.11.G92 1 122 0 92 159yes; 81 1 144 5 225151212.613.87101516yes; 62 1 82 5 102no; 72 1 152 u 162Each polygon circumscribes a circle. What is the perimeter of each polygon?2 in.14.To start, find the length of eachunknown segment.15 in.152 1 15 1 uu3 1 16 1 u16131uP52116 in.3 in.15.9 cm72 in.16.92 cm12 ft140 ft39 ft22 cm19 ft15 cm5 in.17. (B is inscribed in a triangle, whichhas a perimeter of 76 in. What isthe value of x? 8 in.B25 in.xH29 18. Reasoning GHI is a triangle. How can youprove that HI is tangent to (G?mlG 1 mlI 5 90. By the Triangle Angle-Sum Thm.,mlH 5 90, so HI is tangent to (G by Thm. 12-2.Prentice Hall Foundations Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.661 GI

NameClassDateStandardized Test Prep12-1Tangent LinesMultiple ChoiceFor Exercises 1–5, choose the correct letter.A1. AB and BC are tangents to (P. What is the value of x? B73117107146x P73 BC2. Earth’s radius is about 4000 mi. To the nearest mile, what is thedistance a person can see on a clear day from an airplane 5 mi above Earth? G63 mi200 mi4000 mi5660 miY3. YZ is a tangent to (X , and X is the center of the circle.What is the length of the radius of the circle? B412612.8AFE12 cmC5. (A is inscribed in a quadrilateral. What is theperimeter of the quadrilateral? B25 mm60 mm50 mm150 mm3 mm1 cmB10 cm3 cmEG6 cmBFCDF4 cmZ10X4. The radius of (G is 4 cm. Which is atangent of (G? IAB8D12 mmA7.5 mm2.5 mmShort Response6. Given: GI is a tangent to (J .Prove: nFGH nFIH [2] Statements: 1) GI is a tangent to (J; 2) FH ' GI;42 FJGH3) lFHG and lFHI are right '; 4) lFHG O lFHI;5) FH O FH; 6) mlGFH 5 42, mlFIH 5 48; 7) mlHFI 5 42;8) lHFI O lHFG; 9) kFGH O kFIH; Reasons: 1) Given;2) Thm. 12-1; 3) Def. of '; 4) Def. of O; 5) Refl. Prop.;6) Given; 7) Triangle Angle-Sum Thm.; 8) Def. of O;48 9) ASA Post. [1] proof missing steps [0] incorrect orI missing proofPrentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.7

Name12-1ClassDateEnrichmentTangent LinesInscribed Circles and Right TrianglesThe following theorem is about a relationship between the radius of the inscribedcircle of a right triangle and the lengths of the triangle’s sides.BComplete the proof.PGiven: Right nABC with right angle at C. (O is inscribedin nABC. M, N, and P are the points of tangency.Radius ON is drawn.AProve: 2 ON 5 AC 1 BC 2 ABStatementsMONCReasons1) In right nABC, /C is a right angle. (O isinscribed in ABC. M, N, and P are points oftangency. Radius ON is drawn.1) 9 Given2) Draw radii OM and OP.2) 9 Two points determine a line4) ON OMsegment.3) 9 Theorem 12-3 (Two tangentsdrawn to a circle from a pointoutside the circle are congruent.)4) 9 Radii of a circle are congruent.5) /OMC is a right angle.5) 9 Theorem 12-1 (A radius and a6) /ONC is a right angle.6) 9 Theorem 12-1 (A radius and a7) Quad. OMCN is a square.7) 9 Definition of a square8) ON OM CM CN8) 9 The sides of a square are3) BP BM ; CM CN ; AP ANtangent line drawn to the samepoint of contact form a right angle.)tangent line drawn to the samepoint of contact form a rightangle.)congruent.9) 9 Addition Property9) BP 1 CM 5 BM 1 CNAP 1 CM 5 AN 1 CN10) AN 1 CN 5 ACBM 1 CM 5 BCBP 1 AP 5 AB10) 9 Segment Addition Postulate11) BP 1 CM 5 BC11) 9 Substitution Property12) (BP 1 AP) 1 2 CM 5 BC 1 AC13) AB 1 2 ON 5 BC 1 AC12) 9 Substitution Property13) 9 Substitution Property14) 2 ON 5 AC 1 BC 2 AB14) 9 Subtraction PropertyAddition Property andPrentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.8

NameClassDateReteaching12-1Tangent LinesAA tangent is a line that touches a circle at exactly one point.In the diagram, AB is tangent to (Q. You can apply theoremsabout tangents to solve problems.RBQTheorem 12-1If a line is tangent to a circle, then that line forms a right anglewith the radius at the point where the line touches the circle.Theorem 12-2If a line in the same plane as a circle is perpendicular to a radius at its endpoint onthe circle, then the line is tangent to the circle.12HProblemr 68 rKUse the diagram at the right to solve the problems below.GH is tangent to (K .What is the measure of /G?G9What is the length of the radius?You can use the PythagoreanTheorem to find missing lengths.Because GH is tangent to (K , itforms a right angle with the radius.HK2 1 HG2 5 GK2The sum of the angles of a triangleis always 180. Write an equationto find m/G.r2 1 122 5 (9 1 r)2r2 1 144 5 (9 1 r)(9 1 r)m/G 1 m/H 1 m/K 5 180r2 1 144 5 81 1 18r 1 r2m/G 1 90 1 68 5 18063 5 18rm/G 1 158 5 1803.5 5 rm/G 5 22So, the measure of /G is 22 and the length of the radius is 3.5 units.ExercisesIn each circle, what is the value of x?621.O312.x O59 573.33 x 28 Prentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.9x O

NameClassDateReteaching (continued)12-1Tangent LinesIn each circle, what is the value of r?4.rOr5.43159r1O6.88rrO8.2514rTheorem 12-3AIf two segments are tangent to a circle from the same pointoutside the circle, then the two segments are equal in length.DIn the diagram, AB and BC are both tangent to (D. So, theyare also congruent.CAWhen circles are drawn inside a polygon so that the sidesof the polygon are tangents, the circle is inscribed in thefigure. You can apply Theorem 12-3 to find theperimeter, or distance around the polygon.9 in.ZDProblemWB 5 BX 5 5CY 5 XC 5 2YD 5 DZ 5 3M3 in.W5 in.BYXC2 in.(M is inscribed in quadrilateral ABCD.What is the perimeter of ABCD?ZA 5 AW 5 9BNow add to find the length of each side:AB 5 AW 1 WB 5 9 1 5 5 14BC 5 BX 1 CX 5 5 1 2 5 7CD 5 CY 1 YD 5 2 1 3 5 514 1 7 1 5 1 12 5 38DA 5 DZ 1 ZA 5 3 1 9 5 12The perimeter is 38 in.ExercisesEach polygon circumscribes a circle. What is the perimeter of each polygon?7.6 cm34 cm8. 7.5 ft4 ft35 ft7 cm9.49 cm3.5 cm3 cm8 cm6 ft5.25 cmPrentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.108.75 cm

NameClass12-2DateAdditional Vocabulary SupportChords and ArcsChoose the word from the list below that best matches each sentence.arcbisectcentral anglechorddiameterradius1. A segment with endpoints on a circle.chord2. To divide exactly in half.bisect3. A segment from the center of a circle to any point on the circle.radiuscentral angle4. An angle whose vertex is the center of a circle.5. A segment with endpoints on a circle that passes through the center.diameterUse a word from the list above that best describes each central angleMultiple Choice12. The diagram at the right shows a sector of a circle. Which ofthe following defines the boundary of a sector? Ctwo radii and a chordtwo radii and an arctwo diameterstwo chords13. The radius of a circle is 5 in. long. How long is the diameter? I2.5 in.5 in.7.5 in.Prentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.1110 in.

Name12-2ClassDateThink About a PlanChords and Arcs(A and (B are congruent. CD is a chord of both circles.If AB 5 8 in. and CD 5 6 in., how long is a radius?1. Draw the radius of each circle that includes point C. What isCAxBDthe name of each of the two line segments drawn?AC , BC2. Label the intersection of CD and AB point X.3. You know that CD CD. How can you use the converse of Theorem 12-7 toshow that AX 5 XB?Because congruent chords are equidistant from the centers of congruent circles, X is thesame distance from A as it is from B. So, AX 5 XB.4. How long is XB? 4 in.5. Draw in radius BD. What is true about BC and BD? Explain.BC 5 BD; all radii of a circle have the same length.6. Because AD 5 AC 5 BD 5 BC, ACBD is a rhombus.diagonals AB and CD are perpendicularand its7. What can you say about the diagram using Theorem 12-8? AB bisects CD.8. How long is CX ? 3 in.9. How can you use the Pythagorean Theorem to find BC?a2 1 b2 5 c2 ; if CX 5 a, XB 5 b, and CB 5 c, then CB 5 "32 1 42 5 "25 5 5 in.10. How long is the radius of each circle? 5 in.Prentice Hall Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.12

NameClass12-2DatePracticeForm GChords and ArcsIn Exercises 1 and 2, the (X O (E. What can you conclude?Q1.PA2.BXXEDRCSlQXP O lRXS O lAEB O lDEC ; all radii arecongruent; all chords drawn are congruent.Find the value of x.3.3x816FEYlWXY O lDEF ; WY O DF ; all radiiare congruent.54.DWx5.1212.64.9x356.36. In (X , AC is a diameter and ED EB. What can you conclude00about DC and CB ? Explain.ABE0 0DC O CB ; because ED O EB and XB O XD, AC must bea perpendicular bisector of DB by the Converse of thePerpendicular Bisector Theorem. This means DC O CB, so0 0by Theorem 12-6, DC O CB .XCD7. In (D, ZX is the diameter of the circle and ZX ' WY .WWhat conclusions can you make? Justify your answer.WD O0DY because ZX is a perpendicular bisector, and0WX O XY because of Theorem 12-8.DZYFind the value of x to the nearest tenth.5.78.14 x18X6.59.x4.225.410.x1581011. In the figure at the right, sphere O with radius 15 mm isintersected by a plane 3 mm from the center. To thenearest tenth, find the radius of the cross section (Y .3 mm O14.7 mmYPrentice Hall Gold Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.1315 mm

Name12-2ClassDatePractice (continued)Form GChords and Arcs00012. Given: (J with diameter HK ; KL LM MKKProve: nKIL nKIM0 0Statements: 1) KI O KI; 2) KL O KM ; 3) KM O KL; 4) JM O JL; 5) KHis the ' bis. of ML; 6) IM O IL; 7) kKIL O kKLM; Reasons: 1) Refl.Prop. of O; 2) Given; 3) Converse Thm. 12-6; 4) All radii in a circleare O; 5) Converse of ' Bis. Thm.; 6) Def. of a bis.; 7) SSSI JMLH13. Given: AC and DB are diameters of (E.AProve: nEAD nECBStatements: 1) AC and DB are diameters of (E; 2) AE O CE andDE O BE; 3) lAED O lCEB; 4) kEAD O kECB;Reasons: 1) Given; 2) Def. of radius; 3) Vert. Angles are O; 4) SASEDBC(N and (O are congruent. PQ is a chord of both circles.14. If NO 5 12 in. and PQ 5 8 in., how long is the radius to thenearest tenth of an inch? 7.2 in.PN15. If NO 5 30 mm and radius 5 16 mm, how long is PQ to thenearest tenth of a millimeter? 11.1 mmOQ16. If radius 5 12 m and PQ 5 9 m, how long is NO to the nearest tenth? 22.2 m17. Draw a Diagram A student draws (X with a diameter of 12 cm. Inside thecircle she inscribes equilateral nABC so that AB, BC, and CA are all chordsof the circle. The diameter of (X bisects AB. The section of the diameter fromthe center of the circle to where it bisects AB is 3 cm. To the nearest wholenumber, what is the perimeter of the equilateral triangle inscribed in (X ? 31 cm18. Two concentric circles have radii of 6 mm and 12 mm. A segment tangentto the smaller circle is a chord of the larger circle. What is the length of thesegment to the nearest tenth. 20.8 mmPrentice Hall Gold Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.14

NameClass12-2DatePracticeForm KChords and ArcsIn Exercises 1 and 2, the circles are congruent. What can you conclude?A1.HFTo start, look at the chords. If they areequidistant from the center of thecircle, what can be concluded?BGThe chords must be9. congruent0 0FH O AC ; GF O GH O BA O BC ; FH O AC ; lFGH O lABC2.QJKGSBQR TS 9 9RKJ; LMMLTC0QR 9 9 9/QGR 9 9 9lSGT ; lJBK ; lMBL00 0KJ ; ST ; LM/QGS 9 9 90QS 9 9 900 0TR ; KL ; MJlRGT ; lJBM; lKBLFind the value of x.3.3015134.899x85.1812.5x44x256. Reasoning /QRS and /TRV are vertical angles inscribed in (R.00What must be true of QS and TV ? Explain.They must be O because vertical ' are O, and the arcs of O central' in the samecircle are O.Draw a Diagram Tell whether the statement is always, sometimes, ornever true.007. XY and RS are in congruent circles. Central /XZY and central /RTSare congruent. sometimes8. (I (K . The length of chord GH in (I is 3 in. and the length of chord LMin (K is 3 in. /GIH /LKM . always9. /STU and /RMO are central angles in congruent circles. m/STU 5 50 and0 0m/RMO 5 55. SU RO . neverPrentice Hall Foundations Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.15

NameClassDatePractice (continued)12-2Form KChords and ArcsQ10. In the diagram at the right, ST is a diameter of the circleand ST ' QR. What conclusions can you make?0 0 0 0QT O TR , SQ O SR , and QU O UR.S11. In the diagram at the right, IJ is a perpendicularGIbisector of chord GH . What can you conclude?HFind the value of x to the nearest tenth.4.512.15x13.420xJTo start, since the radius isperpendicular to the chord,the chord is bisected.The longer leg of the triangle is 12 4 2 51210.8RKAnswers may vary. Sample: IJ contains thecenter of the circle.TU13.414.10.915.966 .u10xx24(D and (E are congruent. GH is a chord of both circles. Round all answers tothe nearest tenth.16. If DE 5 10 in. and GH 5 4 in., how long is a radius? 5.4 in.GD17. If DE 5 22 cm and radius 5 14 cm, how long is GH ? 17.3 cmEH18. If the radius 5 18 ft and GH 5 32 ft, how long is DE? 16.5 ft19. In the figure at the right, Sphere Z with radius 9 in. isintersected by a plane 4 in. from center Z. To the nearesttenth, find the radius of the cross section (X . 8.1 in.4 in.ZXPrentice Hall Foundations Geometry Teaching ResourcesCopyright by Pearson Education, Inc., or its affiliates. All Rights Reserved.169 in.

NameClassDateStandardized Test Prep12-2Chords and ArcsMultiple ChoiceFor Exercises 1–5, choose the correct letter.C1. The circles at the right are cong

92 1 122 0 92 12. 13. Each polygon circumscribes a circle. What is the perimeter of each polygon? 14. To start, fi nd the length of each unknown segment. P 5 2 1 u1 15 1 u 1 3 1 u1 16 1 u 15. 16. 17. (B is inscribed in a triangle, which has a perimeter of 76 in. What is the value of x? 18. Reasoning GHI is a triangle. How can you prove that HI .

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