AMSCO'S Geometry. New York - Rye High School

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612 Locus and ConstructionExercisesWriting About Mathematics1. Are all of the points that are equidistant from the endpoints of a line segment that is 8 centimeterslong 4 centimeters from the endpoints? Explain your answer.2. What line segment do we measure to find the distance from a point to a line or to a ray?Developing Skills3. What is the locus of points that are 10 centimeters from a given point?4. What is the locus of points equidistant from two points, A and B, that are 8 meters apart?5. What is the locus of points equidistant from two parallel lines 8 meters apart?6. What is the locus of points 4 inches away from a given line, AB g?7. What is the locus of points 3 inches from each of two parallel lines that are 6 inches apart?8. What is the locus of points that are equidistant from two opposite sides of a square?9. What is the locus of points that are equidistant from the vertices of two opposite angles of asquare?10. What is the locus of points that are equidistant from the four vertices of a square? (A locuscan consist of a single point or no points.)11. What is the locus of points in the interior of a circle whose radius measures 3 inches if thepoints are 2 inches from the circle?12. What is the locus of points in the exterior of a circle whose radius measures 3 inches if thepoints are 2 inches from the circle?13. What is the locus of points 2 inches from a circle whose radius measures 3 inches?14. Circle O has a radius of length r, and it is given that r m.a. What is the locus of points in the exterior of circle O and at a distance m from the circle?b. What is the locus of points in the interior of circle O and at a distance m from the circle?c. What is the locus of points at a distance m from circle O?15. Concentric circles have the same center. What is the locus of points equidistant from twoconcentric circles whose radii measure 10 centimeters and 18 centimeters?16. A series of isosceles triangles are drawn, each of which has a fixed segment, AB, as its base.What is the locus of the vertices of the vertex angles of all such isosceles triangles?17. Triangle ABC is drawn with a fixed base, AB, and an altitude to AB whose measure is 3feet. What is the locus of points that can indicate vertex C in all such triangles?
Five Fundamental Loci 613Applying Skills18. What is the locus of the tip of the hour hand of a clock during a 12-hour period?19. What is the locus of the center of a train wheel that is moving along a straight, level track?20. What is the locus of the path of a car that is being driven down a street equidistant from thetwo opposite parallel curbs?21. A dog is tied to a stake by a rope 6 meters long. What is the boundary of the surface overwhich he can move?22. A boy walks through an open field that is bounded on two sides by straight, intersectingroads. He walks so that he is always equidistant from the two intersecting roads. Determinehis path.23. There are two stationary floats on a lake. A girl swims so that she is always equidistant fromboth floats. Determine her path.24. A dime is rolled along a horizontal line so that the dime always touches the line. What is thelocus of the center of the dime?25. The outer edge of circular track is 40 feet from a central point. The track is 10 feet wide.What is path of a runner who runs on the track, 2 feet from the inner edge of the track?14-3 FIVE FUNDAMENTAL LOCIThere are five fundamental loci, each based on a different set of conditions. Ineach of the following, a condition is stated, and the locus that fits the conditionis described in words and drawn below. Each of these loci has been shown as aconstruction in preceding sections.1. Equidistant from two points: Find points equidistant from points A and B.Locus: The locus of points equidistant from twofixed points is the perpendicular bisector of the segmentdetermined by the two points.2. Equidistant from two intersecting lines: Find points equidistant from theintersecting lines AB gand CD g.Locus: The locus of points equidistant from two intersectinglines is a pair of lines that bisect the anglesformed by the intersecting lines.A DALocusLocusBCLocusB
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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
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Chapter Summary 527CHAPTER SUMMARYD
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Review Exercises 529FormulasIn the
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14. The length of a side of a rhomb
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Cumulative Review 5334. The measure
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CHAPTERGEOMETRYOF THECIRCLEEarly ge
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Arcs and Angles 537In the diagram,
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If (O > (Or and mCDX 5 mCrDr 60, t
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Arcs and Angles 541b. mACX mAOC 75
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Arcs and Chords 54313-2 ARCS AND CH
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Arcs and Chords 545ProofFirst draw
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Arcs and Chords 547Since a diameter
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Arcs and Chords 549(5) Since AB CD
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Arcs and Chords 551EXAMPLE 3ProofPr
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Inscribed Angles and Their Measures
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. m/A 5 X 12 mBC c. m/C 5 180 2 (m/
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Inscribed Angles and Their Measures
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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 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: The Meaning of Locus 611Note that i
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 641 and 642: Review Exercises 631VOCABULARY14-2
Page 643 and 644: CUMULATIVE REVIEW Chapters 1-14Part
Page 645 and 646: Part IVAnswer all questions in this
Page 647 and 648: Index 637Biconditional(s), 69-73app
Page 649 and 650: Index 639multiplicative, 5Identity
Page 651 and 652: Index 641Projectionof point on a li
Page 653: Index 643Trichotomy postulate, 264-
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AMSCO'S Geometry. New York - Rye High School

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