AMSCO'S Geometry. New York - Rye High School

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578 <strong>Geometry</strong> of the CirclePSolution Since ST TR SR 9, let ST x and TR 9 x.R(PT)(TQ) 5 (RT)(TS)(2)(10) 5 x(9 2 x)2T10O20 5 9x 2 x 2Qx 2 2 9x 1 20 5 09S(x 2 5)(x 2 4) 5 0x 2 5 5 0 x 2 4 5 0x 5 5 x 5 49 2 x 5 4 9 2 x 5 5Answer ST 5 and TR 4, or ST 4 and TR 5EXAMPLE 3Find the length of a chord that is 20 centimeters from the center of a circle if thelength of the radius of the circle is 25 centimeters.Solution Draw diameter COD perpendicular to chord AB at E.Then OE is the distance from the center of the circle tothe chord.OE 20 DE 5 OD 1 OE CE 5 OC 2 OE5 25 1 205 25 2 205 455 5A diameter perpendicular to a chord bisects the chord. Therefore, AE EB.Let AE EB x.(AE)(EB) 5 (DE)(CE)(x)(x) 5 (45)(5)x 2 5 225x 5 15 (Use the positive square root.)Therefore,AB 5 AE 1 EB5 x 1 x5 30 cm AnswerA5 cm20 cmO25 cmCED25 cmB
Measures of Tangent Segments, Chords, and Secant Segments 579SUMMARYType of Segment Length ExampleFormed by TwoIntersectingChordsIf two chords intersect, theproduct of the measures of thesegments of one chord is equalto the product of the measures ofof the segments of the other.DAOECBFormed by aTangentIntersectinga SecantFormed by TwoIntersectingSecantsIf a tangent and a secant are drawnto a circle from an external point,then the square of the length ofthe tangent segment is equal tothe product of the lengths of thesecant segment and its externalsegment.If two secant segments aredrawn to a circle from anexternal point, then the productof the lengths of one secantsegment and its external segmentis equal to the product of thelengths of the other secantsegment and its external segment.(AE)(EB) 5 (CE)(ED)AOCBP(PA) 2 5 (PC)(PB)PACB OD(PB)(PA) 5 (PD)(PC)ExercisesWriting About Mathematics1. The length of chord AB in circle O is 24. Vanessa said that any chord of circle O that intersectsAB at its midpoint, M, is separated by M into two segments such that the product ofthe lengths of the segments is 144. Do you agree with Vanessa? Justify your answer.2. Secants ABP and CDP are drawn to circle O. If AP CP, is BP DP? Justify youranswer.
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AMSCO’SGEOMETRYAnn Xavier Gantert
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PREFACE✔✔✔✔✔✔Geometry i
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CONTENTSChapter 1ESSENTIALS OF GEOM
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viiiCONTENTSReview Exercises 323Cum
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xCONTENTS14-3 Five Fundamental Loci
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2 Essentials of Geometry1-1 UNDEFIN
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4 Essentials of GeometryThe number
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6 Essentials of GeometryEXAMPLE 1In
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8 Essentials of GeometryRecall that
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10 Essentials of GeometryEXAMPLE 1O
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12 Essentials of GeometryOn the num
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14 Essentials of Geometry7. A, B, C
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16 Essentials of Geometry1. By a ca
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18 Essentials of GeometrySolutiona.
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20 Essentials of GeometryBisector o
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22 Essentials of GeometryIn 11-13,
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24 Essentials of GeometryIncluded S
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26 Essentials of GeometryRight Tria
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28 Essentials of GeometryIn 5 and 6
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30 Essentials of GeometryPR S TProp
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32 Essentials of Geometry12. In iso
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CHAPTER2CHAPTERTABLE OF CONTENTS2-1
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36 LogicSome sentences are true for
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38 LogicNegationsIn the study of lo
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40 Logic2. a. Give an example of a
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42 Logic2-2 CONJUNCTIONSWe have ide
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44 LogicA compound sentence may con
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46 LogicEXAMPLE 4Three sentences ar
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48 Logic31. I have the hiccups and
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50 LogicAnswers(1) k ∨ q a. Every
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52 Logic8. A gram is a measure of l
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54 LogicParts of a ConditionalThe p
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56 Logic2. “In order to succeed,
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58 LogicExercisesWriting About Math
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60 Logic42. The conditional p → q
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62 Logicp: Angle A and B are congru
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64 LogicOn Friday, p is true and q
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66 LogicConditional If Jessica like
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68 LogicIn 7-10: a. Write the inver
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70 LogicRecall that a conjunction i
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72 LogicEXAMPLE 2The statement “I
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74 LogicApplying Skills17. Let p re
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76 LogicThe Law of Disjunctive Infe
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78 Logic(1) Eliminate the second ro
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80 Logic17. If 2b 6 14, then 2b
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82 Logicglass” is also true. Appl
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84 Logic7. When p → q and p ∧ q
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86 Logic•A hypothesis, also calle
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88 Logic8. July is a winter month i
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90 LogicExploration1. A tautology i
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92 Logic11. The first two sentences
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94 Proving Statements in Geometry3-
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96 Proving Statements in GeometryDe
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98 Proving Statements in GeometryBe
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100 Proving Statements in Geometry1
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102 Proving Statements in GeometryE
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104 Proving Statements in GeometryA
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108 Proving Statements in GeometryC
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110 Proving Statements in GeometryS
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116 Proving Statements in GeometryF
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120 Proving Statements in GeometryJ
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128 Proving Statements in GeometryC
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CHAPTER1344CHAPTERTABLE OF CONTENTS
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136 Congruence of Line Segments,Ang
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176 Congruence Based on TrianglesAC
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178 Congruence Based on Triangles6.
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180 Congruence Based on TrianglesDe
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182 Congruence Based on TrianglesTh
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184 Congruence Based on Triangles(C
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186 Congruence Based on Triangles5-
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196 Congruence Based on TrianglesAp
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198 Congruence Based on TrianglesCo
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200 Congruence Based on TrianglesCo
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202 Congruence Based on TrianglesEX
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204 Congruence Based on TrianglesCH
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206 Congruence Based on TrianglesCU
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208 Congruence Based on Triangles13
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210 Transformations and the Coordin
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CHAPTER7CHAPTERTABLE OF CONTENTS7-1
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264 Geometric InequalitiesThis corr
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266 Geometric InequalitiesExercises
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268 Geometric InequalitiesSubtracti
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270 Geometric Inequalities7-3 INEQU
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274 Geometric InequalitiesEXAMPLE 2
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276 Geometric InequalitiesHands-On
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278 Geometric InequalitiesABMCProof
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282 Geometric InequalitiesGivenTo p
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284 Geometric InequalitiesCIf the m
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286 Geometric Inequalities7.3 If th
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288 Geometric InequalitiesCUMULATIV
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292 Slopes and Equations of LinesAl
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Developing SkillsIn 3-11, in each c
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y 2 bx 2 a 5 m The Equation of a Li
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5 ? 3 2(2) 1 1 5 ? 2(0) 1 7 5 ? 2
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zontal line, they all have the same
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Midpoint of a Line Segment 303Proof
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Midpoint of a Line Segment 305EXAMP
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The Slopes of Perpendicular Lines 3
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The Slopes of Perpendicular Lines 3
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Developing SkillsIn 3-12: a. Find t
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Coordinate Proof 3138-5 COORDINATE
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Coordinate Proof 315ProofThis is a
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Concurrence of the Altitudes of a T
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VOCABULARY8-1 Slope • x • y8-2
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Cumulative Review 325CUMULATIVE REV
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Cumulative Review 327Part IVAnswer
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Proving Lines Parallel 3299-1 PROVI
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Proving Lines Parallel 331In the di
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Proving Lines Parallel 333The proof
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Properties of Parallel Lines 3359-2
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Properties of Parallel Lines 337The
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gEJ. Therefore, AB g CD gbecause i
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c. Is the measure of an exterior an
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Parallel Lines in the Coordinate Pl
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Parallel Lines in the Coordinate Pl
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. Find the midpoints of two sides o
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The Sum of the Measures of the Angl
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The Sum of the Measures of the Angl
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Proving Triangles Congruent by Angl
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Proving Triangles Congruent by Angl
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The Converse of the Isosceles Trian
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The Converse of the Isosceles Trian
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In 3-6, in each case the degree mea
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(1) Since any line segment may beex
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Proving Right Triangles Congruent b
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Interior and Exterior Angles of Pol
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Chapter Summary 373CHAPTER SUMMARYD
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AB g In 1-5, CD gand these lines a
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2. The statement “If two angles f
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CHAPTERQUADRILATERALSEuclid’s fif
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The Parallelogram 381ADBCQuadrilate
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The Parallelogram 383DEFINITIONThe
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Proving That a Quadrilateral Is a P
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Proving That a Quadrilateral Is a P
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The Rectangle 38910-4 THE RECTANGLE
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The Rectangle 391EXAMPLE 1Given: AB
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The Rhombus 39316. The coordinates
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The Rhombus 395Theorem 10.15If a qu
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2. Concepta said that if the length
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The Square 39910-6 THE SQUAREDEFINI
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The Square 401EXAMPLE 2In square PQ
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The Trapezoid 403The Isosceles Trap
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The Trapezoid 405Proof We will show
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The Trapezoid 407The slopes of BC a
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18. Prove Theorem 10.23, “The len
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Areas of Polygons 4115. Find the ar
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Chapter Summary 413PostulateTheorem
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Review Exercises 41511. The diagona
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Cumulative Review 417CUMULATIVE REV
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CHAPTERTHEGEOMETRYOF THREEDIMENSION
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Points, Lines, and Planes 421Theore
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Perpendicular Lines and Planes 4231
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Perpendicular Lines and Planes 425G
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Perpendicular Lines and Planes 427l
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Perpendicular Lines and Planes 429G
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gPlanes p and q intersect in AB. In
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16. Prove that if two points are ea
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Parallel Lines and Planes 435Theore
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BF intersecting q at . Since p and
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ProofTwo lines perpendicular to the
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Since the bases are parallel, the c
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Surface Area of a Prism 443EXAMPLE
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10. A prism has bases that are trap
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Volume of a Prism 447The other pris
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Pyramids 44911-6 PYRAMIDSA pyramid
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Pyramids 451For example, consider a
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Cylinders 45314. Let F be the verte
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Express this result as a rational a
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Cones 457We can make a model of a r
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7. The volume of a cone is 127 cubi
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Spheres 461Proof Statements Reasons
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Spheres 463When we round to the nea
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Chapter Summary 465• Perpendicula
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Vocabulary 46711.14 The intersectio
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Review Exercises 46914. A prism and
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Cumulative Review 471b. Using the s
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Cumulative Review 47314. Line segme
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Ratio and Proportion 47512-1 RATIO
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Ratio and Proportion 477ProofWe can
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Ratio and Proportion 479EXAMPLE 3Th
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Given ABC, D is the midpoint of AC,
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Proportions Involving Line Segments
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Proportions Involving Line Segments
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If two polygons are similar, then t
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Proving Triangles Similar 4899. Tri
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Proving Triangles Similar 491We can
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Proving Triangles Similar 493EXAMPL
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Dilations 49511. If CA 48, DA 12,
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Dilations 497Slope ofAB 5 d b 2 2 a
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EXAMPLE 2SolutionDilations 499Find
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Dilations 50129. The vertices of oc
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Proportional Relations Among Segmen
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Proportional Relations Among Segmen
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Concurrence of the Medians of a Tri
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(2) Find the equation of AM g and t
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Proportions in a Right Triangle 511
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Proportions in a Right Triangle 513
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Pythagorean Theorem 51512-9 PYTHAGO
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Pythagorean Theorem 517EXAMPLE 2Whe
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Pythagorean Theorem 519EXAMPLE 3In
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The Distance Formula 52123. A young
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The lengths of the sides of the qua
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Developing SkillsIn 3-10, the coord
Page 537 and 538: Chapter Summary 527CHAPTER SUMMARYD
Page 539 and 540: Review Exercises 529FormulasIn the
Page 541 and 542: 14. The length of a side of a rhomb
Page 543 and 544: Cumulative Review 5334. The measure
Page 545 and 546: CHAPTERGEOMETRYOF THECIRCLEEarly ge
Page 547 and 548: Arcs and Angles 537In the diagram,
Page 549 and 550: If (O > (Or and mCDX 5 mCrDr 60, t
Page 551 and 552: Arcs and Angles 541b. mACX mAOC 75
Page 553 and 554: Arcs and Chords 54313-2 ARCS AND CH
Page 555 and 556: Arcs and Chords 545ProofFirst draw
Page 557 and 558: Arcs and Chords 547Since a diameter
Page 559 and 560: Arcs and Chords 549(5) Since AB CD
Page 561 and 562: Arcs and Chords 551EXAMPLE 3ProofPr
Page 563 and 564: Inscribed Angles and Their Measures
Page 565 and 566: . m/A 5 X 12 mBC c. m/C 5 180 2 (m/
Page 567 and 568: Inscribed Angles and Their Measures
Page 569 and 570: Tangents and Secants 559Theorem 13.
Page 571 and 572: ProofTangents and Secants 561We wil
Page 573 and 574: Tangents and Secants 563The proofs
Page 575 and 576: Developing SkillsIn 3 and 4, ABC is
Page 577 and 578: Hands-On ActivityConsider any regul
Page 579 and 580: Angles Formed by Tangents, Chords,
Page 585 and 586: Measures of Tangent Segments, Chord
Page 587: Measures of Tangent Segments, Chord
Page 591 and 592: Circles in the Coordinate Plane 581
Page 593 and 594: T 3, 22lation :Circles in the Coord
Page 595 and 596: Developing SkillsIn 3-8, write an e
Page 597 and 598: Circles in the Coordinate Plane 587
Page 599 and 600: How to Proceed(1) Solve the pair of
Page 601 and 602: Tangents and Secants in the Coordin
Page 603 and 604: Chapter Summary 59321. a. Write an
Page 605 and 606: Chapter Summary 59513.12 The measur
Page 607 and 608: Vocabulary 597Formulas(Continued)Ty
Page 609 and 610: Review Exercises 59911. Two circles
Page 611 and 612: Cumulative Review 6012. The coordin
Page 613 and 614: Find the coordinatesR 90+ r y5x.Cum
Page 615 and 616: Constructing Parallel Lines 60514-1
Page 617 and 618: Constructing Parallel Lines 607Exer
Page 619 and 620: PL h b. Choose point N on . Constru
Page 621 and 622: The Meaning of Locus 611Note that i
Page 623 and 624: Five Fundamental Loci 613Applying S
Page 625 and 626: SolutionAnswerBCAB g CD g . The lo
Page 627 and 628: Points at a Fixed Distance in Coord
Page 629 and 630: Equidistant Lines in Coordinate Geo
Page 635 and 636: Points Equidistant from a Point and
Page 637 and 638: Points Equidistant from a Point and
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Points Equidistant from a Point and
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Review Exercises 631VOCABULARY14-2
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CUMULATIVE REVIEW Chapters 1-14Part
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Part IVAnswer all questions in this
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Index 637Biconditional(s), 69-73app
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Index 639multiplicative, 5Identity
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Index 641Projectionof point on a li
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Index 643Trichotomy postulate, 264-
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AMSCO'S Geometry. New York - Rye High School

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