12-1 Tangent Lines
12-112-1Tangent Lines1. PlanObjectives12To use the relationshipbetween a radius anda tangentTo use the relationshipbetween two tangents fromone pointExamples12345Finding Angle MeasuresReal-World ConnectionFinding a TangentUsing Theorems 12-3Circles Inscribed in PolygonsWhat You’ll LearnCheck Skills You’ll Need To use the relationshipFind each product.between a radius and atangent To use the relationshipbetween two tangents fromone point. . . And WhyGO for Help Skills Handbook page 754 and Lesson 8-11. (p 3)2 p2 6p 9w2 20w 1002. (w 10)2x 2 Algebra Find the value of x. Leave your answer in simplest radical form.A4.5.8"5xTo find the distance betweenthe centers of two dirt bikegears, as in Example 2m2 – 4m 43. (m - 2)216817B20xAC6. 12C 2"3013xBABC16New Vocabulary tangent to a circle point of tangency inscribed in circumscribed aboutMath BackgroundA tangent to a circle is related tothe geometric interpretationof a derivative, the fundamentalconcept of differential calculus.A derivative measures the rate ofchange of a function at any pointand corresponds to the slope ofthe tangent to the graph of thefunction at that point.1Using the Radius-Tangent RelationshipIn Chapter 8, you studied the tangent ratio in right triangles. The tangents you willstudy here relate to circles.Vocabulary TipAThe word tangent mayrefer to a line, ray, orsegment.BMore Math Background: p. 660CLesson Planning andResourcesA tangent to a circle is a line in the plane of thecircle that intersects the circle in exactly one point.The point where a circle and a tangent intersectis the point of tangency.SBA is a tangent ray and BA is a tangent segment.Theorem 12-1 relates a tangent and a radius in a given circle. You will write anindirect proof for Theorem 12-1 in Exercise 29.See p. 660E for a list of theresources that support this lesson.Key ConceptsTheorem 12-1AIf a line is tangent to a circle, then the line is perpendicularto the radius drawn to the point of tangency.PowerPoint* )Bell Ringer PracticeAB ' OPPOCheck Skills You’ll NeedBFor intervention, direct students to:Skills Handbook, p. 754The Pythagorean TheoremYou can use Theorem 12-1 to solve problems involving tangents to circles.Lesson 8-1: Example 2Extra Skills, Word Problems, ProofPractice, Ch. 8662Chapter 12 CirclesSpecial NeedsBelow LevelL1Have students draw two intersecting lines and fit aquarter between the lines so that it is tangent to bothlines. Students measure the distances from the pointof intersection to the points of tangency.662learning style: tactileL2Students may be confused by using tangent in a newway. Try to use the phrases tangent of an angle andtangent to a circle to help reinforce the difference.learning style: verbal
145AABBBEDEDEDCCCMultiple Choice ML and MN are tangent to O.Find the value of x.586390117EDCBA3CBA2CBA1DDEE2. TeachFinding Angle MeasuresEXAMPLETest-Taking TipLO 117 Since ML and MN are tangent to O, &L and &Nare right angles. LMNO is a quadrilateral whose anglemeasures have a sum of 360.Remember that youcan find the sum ofthe angles of apolygon with n sidesusing the formula(n – 2)180.Guided Instructionx MConnection to Language ArtsNThe term tangent is derived fromthe Latin verb tangere, whichmeans “to touch.”m&L m&M m&N m&O 36090 x 90 117 360297 x 360x 63Substitute.Simplify.Math TipSolve.For a ray or segment to be tangentto a circle, the line containing theray or segment must be tangentto the circle. Note that a radiusis never tangent to a circle.The correct answer is B.Quick CheckD1 ED is tangent to O. Find the value of x. 52Ex O238 2Real-WorldEXAMPLEConnectionDirt Bikes A dirt bike chain fits tightly aroundtwo gears. The chain and gears form a figurelike the one at the right. Find the distancebetween the centers of the gears.AReal-WorldConnectionThis motorcycle and manyother two-wheeled vehicleshave chain-drive systems likethe one shown in Example 2.9.3 in.2.4 in.C.Bn.26.5 in26.5 iE9.3 in.Label the diagram.DDraw AE parallel to BC .Additional Examples2.4 in.AD 2 AE 2 ED2 26.5 2Make sure that students rememberthe properties of rectangles wellenough to prove that ABCE is arectangle. Ask: How do you knowthat &BAE and &CEA are rightangles? BC n AE, lABC andlBCE are right angles, andsame-side interior angles aresupplementary. How do you knowthat AE BC? Opposite sides ofa rectangle are congruent.PowerPoint1 BA is tangent to C at point A.Find the value of x.ABCE is a rectangle. #AED is a right triangle with AE 26.5 in. andED 9.3 - 2.4 6.9 in.AD 2EXAMPLE 6.92AD 2 749.86AD 27.383572Pythagorean TheoremSubstitute.CSimplify.x AUse a calculator to find the square root.The distance between the centers is about 27.4 in.Quick Check2 A belt fits tightly around two circularpulleys, as shown at the right. Find thedistance between the centers of the pulleys.about 35.5 in.22 35 in.8 in.14 in.BTheorem 12-2 (next page) is the converse of Theorem 12-1. You can use it to provethat a line or segment is tangent to a circle. You can also use it to construct atangent to a circle (see Exercise 24). You will prove this theorem in Exercise 33.682 A belt fits tightly around twocircular pulleys, as shown below.Find the distance between thecenters of the pulleys.X15 cmY7 cm3 cmLesson 12-1 Tangent Lines663OZAdvanced LearnersEnglish Language Learners ELLL4After Example 5, have students write an equation forthe radius of a circle inscribed in an equilateraltriangle with sides of length s.learning style: verbalSome students may confuse the terms circumscribeand inscribe. It helps students to remember that afigure that is circumscribed goes around anotherfigure and a figure that is inscribed is in anotherfigure.learning style: verbalP15 cmWN 15.5 cm3 O has radius 5. Point P isoutside O such that PO 12,and point A is on O such thatPA 13. Is PA tangent to O at A?Explain. No; PO2 u PA2 OA2.663
Guided InstructionKey ConceptsTheorem 12-2Math TipPoint out that this lesson providesanother way to define an inscribedcircle: A circle is inscribed in apolygon if it is tangent to eachside of the polygon.AB is tangent to O.3Additional Examples247Substitute.L213 If NL 4, LM 7, and NM 8, is ML tangent to N at L? Explain.No; 42 7 2 u 82.Using Multiple TangentsIf a circle is circumscribed about a triangle (Chapter 5), the triangle is inscribed inthe circle. Similarly, when a circle is inscribed in a triangle, as in the diagram, thetriangle is circumscribed about the circle. Each side of the triangleis tangent to the circle. The tangent segments from each vertex arecongruent. You will prove this theorem in Exercise 30.YSKey ConceptsZ6 ftTheorem 12-3The two segments tangent to a circle from a point outsidethe circle are congruent.TUIs kMLN a right triangle?By the Converse of the Pythagorean Theorem, #MLN is a right triangle with right&L. Therefore ML ' NL, and ML is tangent to N at L by Theorem 12-2.Quick CheckCXM25N625 625 Simplify.5 C is inscribed in quadrilateralXYZW. Find the perimeter ofXYZW.11 ft0NM272 242 0 252O8 ftR LM2BFinding a TangentEXAMPLENL2STOIs ML tangent to N at L? Explain.4 QS and QT are tangent to Oat points S and T, respectively. Givea convincing argument that thediagonals of quadrilateral QSOTare perpendicular.QSOT is a kite or a rhombus,so its diagonals areperpendicular.P* )PowerPointQAIf a line in the plane of a circle is perpendicular to a radius atits endpoint on the circle, then the line is tangent to the circle.7 ftAB CBWABOC64 ft4Resources Daily Notetaking Guide 12-1 Daily Notetaking Guide 12-1—L1Adapted InstructionExtend BC and GF to intersect inpoint H. By Theorem 12-3, HC HF,or HB BC HG GF.By Theorem 12-3 again, HB HG,so by the Subtraction Propertyof Equality, BC GF.ClosureFind the radius of the circleinscribed in the right trianglebelow.n.4 in.5i1 in.664Using Theorem 12-3Quick Check664Chapter 12 CirclesCBThe diagram represents a chain drive systemon a bicycle. Give a convincing argumentthat BC GF.L33 in.EXAMPLEGFCBHG4 Critical Thinking Give a convincing argument that BC GF above if you* ) * )know that BC and GF never intersect.44If BC and GF never intersect, then BCFG is a rectangle.F
53. PracticeCircles Inscribed in PolygonsEXAMPLE O is inscribed in #ABC. Find the perimeter of #ABC.AD AF 10 cmBD BE 15 cmCF CE 8 cmABOThe two segments tangent to acircle from a point outside thecircle are congruent.p AB BC CA10 cm D 15 cmF8 cm1 A B 1-4, 6-10, 13-15, 17-19,E24-27, 29CDefinition of perimeter p2 A B AD DB BE EC CF FASegment Addition Postulate 10 15 15 8Substitute. 8 10Assignment Guide5, 11, 12, 16, 20-23,28, 30C Challenge31-33 66The perimeter is 66 cm.Quick CheckEXERCISES15 cm XP5 O is inscribed in #PQR. #PQR has a perimeter of88 cm. Find QY. 12 cmTest PrepMixed ReviewQOYZ17 cm R34-3839-46Homework Quick CheckTo check students’ understandingof key skills and concepts, go overExercises 2, 4, 14, 21, 29.For more exercises, see Extra Skill, Word Problem, and Proof Practice.Practice and Problem SolvingAPractice by Example x 2 Algebra Assume that lines that appear to be tangent are tangent. O is the centerof each circle. Find the value of x.Example 1(page 663)1.2.3. 301204760 forx OHelpO x x O43 60 GOGPS Guided Problem SolvingL3L4EnrichmentL2ReteachingL1Adapted PracticeExamples 2, 4(pages 663, 664)A belt fits snugly around the two circular pulleys shown.PracticeNameClassL3DatePractice 12-1R4. Find the distance between the centers of thepulleys. Round to the nearest hundredth. 14.04 in.ReflectionsyUse the figures to complete Exercises 1–5.S41. For figure IJKL, draw its reflection image in each line.a. x-axisb. y-axis2ⴚ6ⴚ4KIJ6 xOⴚ2ⴚ2RS and QP arecommon tangents.5. Give a convincing argument why the beltlengths RS and QP are equal. See margin.For the pulley system shown, use the lengths givenbelow. Find the missing length to the nearest tenth.6. MQ 10 cm, NP 4 cm,QP 14 cm, MN 7 cm 15.2Example 3(page 664) u8. No;9. Yes; 2.52 62 6.52.10. Yes; 62 82 102.52162M4 in.5 in.14 in.QN152.53. In the diagram, M N O is the image of MNO.a. Name the images of M and N.b. List the pairs of corresponding sides.P4. A B C D is the image of ABCD.a. Name the images of A and C.b. List the pairs of corresponding sides.Exercises ��D7. MQ 5 in., NP 4 in.,QP 20 in., MN 7 in. 20.06.510DCMDetermine whether a tangent line is shown in each diagram. Explain. 8-10.See left.8.9.10.65162.Lⴚ42. In the diagram, C D E F is the image of CDEF.a. Name the images of C and F.b. List the pairs of corresponding sides. Pearson Education, Inc. All rights reserved.Vocabulary Tip5. For figure WXYZ, draw its reflection image in each line.a. x-axisb. y-axisCⴕy4Wⴚ6ⴚ4ⴚ2XOZⴚ2246 xYⴚ4State whether each transformation appears to be an isometry. Explain.6.7.8.Given points T(2, 4), A(–3, –4), and B(0, –4), draw kTAB and its reflectionimage in each line.9. x-axis10. y-axis11. x -312. y 46168Lesson 12-1 Tangent Lines6655. Extend RS and QP untilthey meet at a point, H.By Thm. 11-3, RH QH,or SH RS QP PH.By 11-3 again, SH PH.Thus, RS QP.665
Connection to AstronomyExample 5Exercise 16 As the diagramsuggests, a solar eclipse occurswhen the moon blocks sunlightfrom reaching Earth. A totaleclipse occurs when the mooncompletely blocks the sun’s rays,as in the region between themoon and Earth bounded bythe common internal tangentsto the sun and the moon.Exercise 22 Students may benefitfrom a review of the properties of30 -60 -90 triangles.Tactile LearnersExercise 23 Have studentsmanipulate a nickel, a dime, anda quarter to show how threecircles can be mutually tangent.(page 665)6 cmBApply Your Skills x 2 Algebra Assume that lines that appear to be tangent are tangent. O is the centerof each circle. Find the value of x to the nearest tenth.8 in.P 15 in.13.14.15.10cmP16a. external13Qx5x9in.Ob. external12O x7 cm xc. internalx3.6 cmQ16. Solar Eclipse Common tangents to two circles may be internal or external.If you draw a segment joining the centers of the circles, a common internaltangent will intersect the segment. A common external tangent will not.Real-WorldConnectionThis “diamond ring” effect ina solar eclipse may be seenby a person on Earth at theend of a common externaltangent of the sun and moon.(See diagram at right.)For this cross-sectional diagram ofthe sun, moon, and Earth duringa solar eclipse, use the termsSunEarthabove to describe the typesof tangents of each color. a-c. SeeMoona. redb. blue c. green above.not to scaled. Which tangents show the extent blue lines; green lineson Earth’s surface of total eclipse? Of partial eclipse?e. Reasoning In general, does every pair of circles have common tangents ofboth types? Explain. No; explanations may vary. Sample: Two circlesthat have a common center have no common tangents.hEarth The circle at the right represents Earth. The radius of Earthdis about 6400 km. Find the distance d that a person can see on arclear day from each of the following heights h above Earth.rRound your answer to the nearest tenth of a kilometer.17. 100 m35.8 km18. 500 m19. 1 km113.1 km80.0 km20. BD and CK at the right are diameters of A. 57.5BP and QP are tangents to A. What is m&CDA?GOnlineHomework HelpVisit: PHSchool.comWeb Code: aue-1201666b. Answers may vary.Sample: If you draw2 diagonals for bothsquares, 8 O areformed in the entirefigure with 4 in thesmall square.6663.4 in.3.6 in.9 cm21. History Leonardo da Vinci wrote, “When each ofGPS two squares touch the same circle at four points,one is double the other.” a-b. See margin.a. Sketch a figure that illustrates this statement.b. Writing Explain why the statement is true.21. a.1.9 in.12.14.2 in.16 cm8 cm3.7 in.Exercise 27 Before drawing aExercises 31–33 You may wantstudents to work with partnersor in small groups. Suggest thatthey begin each proof by writinga plan.11.78 cmConnection to CoordinateGeometrytangent segment, students mustfind its length, 4, using thePythagorean Theorem. Then theycan place a compass point at(0, 5), swing an arc of length 4until it intersects the circle, andjoin either point of intersectionand (0, 5).Each polygon circumscribes a circle. Find the perimeter of the polygon.BAC25 KPDQ22. Multiple Choice A regular hexagon iscircumscribed about the ring surrounding theclock face. The diameter of the ring is 10 in.What is the perimeter of the hexagon? C30.0 in.34.6 in.43.3 in.51.7 in.Chapter 12 Circles24.30. a.TR30. 1. BA and BC are tangentto (O at A and C (Given)2. AB # OA and BC #OC (If a line is tan. to acircle, it is # to theradius.) 3. kBAO andkBCO are right . (Def.
23. All four are O; the twotangents to each coinfrom A are O, so bythe Trans. Prop., allare O.23. Critical Thinking A nickel, a dime, and aquarter are touching as shown. Tangentsare drawn from point A to both sides ofeach coin. What can you conclude aboutthe four tangent segments? Explain.EBPowerPointLesson Quiz24. Constructions Draw a circle. Label thecenter T. Locate a point on the circle andlabel it R. Construct a tangent to T at R.See margin.AC is tangent to O at A, and ml1 70.26. 90 –ml4DCPA and PB are tangentto C. Use the figure forExercises 1–3.A25. Find m&4. 35180 2 xQR or x ;221is 2 ml1.4. Assess & ReteachA21 cmO126. Let m&1 x. Find m&4 in terms of x.What is the relationship between &1 and &4?B2C 92.8 3 4A27. Coordinate Geometry Graph the equationx 2 y 2 9. Then draw a segment from (0, 5)tangent to the circle. Find the length of the segment.See back of book.28. Maintenance Mr. Gonzales is replacing a cylindrical airconditioning duct. He estimates the radius of the duct byfolding a ruler to form two 6-in. tangents to the duct. Thetangents form an angle. Mr. Gonzales measures the anglebisector from the vertex to the duct. It is about 2 34 in. long.What is the radius of the duct? about 5.2 in.B1.Find the value of x. 87.22. Find the perimeter ofquadrilateral PACB. 82 cmr33. Find CP. 29 cm6 in.2 4 in.Real-World8 cmOACareers HVAC techniciansoften specialize in eitherinstallation or maintenanceand repair of heating,ventilation, and airconditioning systems.Step 3 The assumption that line n is not perpendicular to OP is f. 9, soline n ' OP.ProofA30. Prove Theorem 12-3.Given: BA and BC are tangent to Oat A and C, respectively.Prove: BA BC See margin.31. Given: BC is tangent to A at D.DB DCProve: AB AC See margin.BC32. Given: A and B with commontangents DF and CESeeProve: #GDC , #GFE margin.DClesson quiz, PHSchool.com, Web Code: aua-1201of rt. k) 4. AO O OC(Radii of a circle are O.)5. BO O BO (Refl. Prop.of O) 6. kBAO O kBCO(HL Thm.) 7. BA O BC(CPCTC)31. 1. BC is tangent to (Aat D. (Given)2. DB O DC (Given)3. AD # BC (If a line istan. to a circle, it is # tothe radius.) 4. lADBEGADJ4 cmB4. Find AB to the nearest tenth.20.4 cm5. What type of specialquadrilateral is AHJB? Explainhow you know. Trapezoid;the tangent line forms rightangles at vertices H and J, soHA n JB. Because HA u JB,AHJB is not a parallelogrambut a trapezoid.Alternative AssessmentOAB20 cmP L KnStep 2 If line n is not perpendicular to OP, some other segment OL must) bea. 9 to line n. By the Ruler Postulate, there is a point K on PL suchthat PK 2PL, so PL b. 9. #OPL #OKL by c. 9, so OP OKbecause d. 9. Since P and K are the same distance from O, both Kand P are on O. This contradicts the given fact that line n is e. 9 to O at P. a. '; b. LK; c. SAS; d. CPCTC; e. tangent; f. falseConnection* )HJ is tangent to A and to B.Use the figure for Exercises 4 and5.HGiven: Line n is tangent to O at P.Step 1 Assume that line n is notperpendicular to OP.20 cmC29. Complete the following indirect proof of Theorem 12-1.Prove: line n ' OPx PCBFLesson 12-1 Tangent Lines667and lADC are rt. '(Def. of #) 5. lADB OlADC (rt. ' are O) 6. ADO AD (Refl. Prop. of O)7. kADB O kADC (SAS)8. AB O AC (CPCTC) Have students use compassand straightedge to constructa circle and then construct aline through a point on thecircle perpendicular to aradius of the circle. Have students repeat theconstruction using a differentradius and then find theintersection of the two linesthey constructed. After constructing each lineand finding the intersection,students should state aconclusion based on atheorem in this lesson.667
Test PrepCChallengeProof33. Write an indirect proof of Theorem 12-2.Given:ResourcesFor additional practice with avariety of test item formats: Standardized Test Prep, p. 711 Test-Taking Strategies, p. 706 Test-Taking Strategies withTransparenciesA* )AB ' OP at P.* )PProve: AB is tangent to O.See back of book.OB34. Two circles that have one point in common are tangent circles. Given anytriangle, explain how to draw three circles that are centered at each vertex ofthe triangle and are tangent to each other.At each vertex, let the radius of a circle be the distance fromthe vertex to either point of tangency of the incircle.Test PrepMultiple Choice[2] 32 3x2x 9x 4.5OR equivalentsolution39.32. 1. (A and (B withcommon tangents DFand CE (Given)2. PG GC andGE GF (Two tan.segments from a pt.to a circle are O.)[1] correct eq.solvedincorrectlyPoint O is the center of each circle. Assume the lines that appear tangent aretangent. What is the value of the variable?35.CO 114 x 937.COx8Short ResponseGDGF3. GC 1, GE 1A.B.C.D.26576611436.FA.B.C.D.89151738.JO4. 12Ox39. Find the value of x. Show your work.See above left.8DF.G.H.J.2345AOGFGEx32PECB(Trans. Prop. of )5. lDGC O lEGF (Vert.' are O.) 6. kGDC MkGFE (SAS M Thm.)222834403(Div. Prop. of )GDGCx 56 F.G.H.J.Mixed ReviewGO forHelpLesson 11-8Two cubes have heights 6 in. and 8 in. Find each ratio.40. similarity ratio41. ratio of surface areas42. ratio of volumes3:49 : 1627 : 64Lesson 8-3 x 2 Algebra Find the value of x. Round answers to the nearest tenth.43.44.5x 15x 845.7.53x 9Lesson 7-228.168.229.1The polygons are similar. (a) State the similarity ratio and (b) find the values ofthe variables. a. 10 : 17a. 4 : 146.b. m 1.82; n 3.7811.9m2.675.4668668Chapter 12 Circles1247.6.5n1.57abc6b. a 1.625; b 1.75; c 3
Theorem 12-1 relates a tangent and a radius in a given circle.You will write an indirect proof for Theorem 12-1 in Exercise 29. You can use Theorem 12-1 to solve problems involving tangents to circles. 12-1 11 Using the Radius-Tangent Relationship p2 6p 9 w 2 20w 100 m –4m 4 2"30
1 SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMS Tangent Problem Definition (Secant Line & Tangent Line) Example 1 Find the equation of the tangent lines to the curve 1 3x 2 y at the
SECANT. Line c intersects the circle in only one point and is called a TANGENT to the circle. a b c TANGENT/RADIUS THEOREMS: 1. Any tangent of a circle is perpendicular to a radius of the circle at their point of intersection. 2. Any pair of tangents drawn a
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
Tangent Pile Wall with Seal Piles. A third variant of the secant pile/tangent pile retaining wall options is a tangent pile wall with “seal” piles placed behind and between the structural tangent piles. T he seal piles do not pro
a. Sine, Cosine and Tangent are _ functions that are related to triangles and angles i. We will discuss more about where they come from later! b. We can evaluate a _, _ or _ just like any other expression c. We have buttons on our calculator for sine, cosine and tangent i. Sine ii. Cosine iii. Tangent d.
Sep 24, 2014 · called concentric. A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the centers of the two circles. Example 3: Tell whether the common tangents are .
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common internal tangent A common external tangent two circles. Notes: intersects the segment that joins the centers of the two circles .
