AMSCO'S Geometry. New York - Rye High School

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308 Slopes and Equations of LinesProof: Let P(x 1, y 1) and Q(x 2, y 2) be two points on line l. Then:slope of l 5 y 2 2 y 1x 2 2 x 1Under a translation T a,b, the images of P and Q have coordinatesPr(x 11 a, y 11 b) and Qr(x 21 a, y 21 b) . Therefore, the slope of l, the imageof l,isslope of lr 5 (y 2 1 b) 2 (y 11 b)(x 2 1 a) 2 (x 1 1 a)5 y 2 2 y 1 1 b 2 bx 2 2 x 1 1 a 2 a5 y 2 2 y 1x 2 2 x 1As a simple application of Theorem 8.2, we can show that the slopes of anytwo perpendicular lines are negative reciprocals of each other:Theorem 8.3aIf two non-vertical lines are perpendicular, then the slope of one is the negativereciprocal of the other.Proof: Let l 1and l 2be two perpendicularlines that intersect at (a, b). Underthe translation (x, y) → (x a, y b),the image of (a, b) is (0, 0).Theorem 8.2 tells us that if theslope of l 1is m, then the slope of itsimage, l 1r, is m. Since l 1and l 2are perpendicular,their images, l 1r and l 2r , arealso perpendicular because translationspreserve angle measure. Usingwhat we established at the beginningof the section, since the slope of l 1r ism, the slope of l is 2 1 2r m. Slope is preservedunder a translation. Therefore,the slope of l 2is 2m1 .y(x 1 a, y 1 b)O(x 2 a, y 2 b)(x 1, y 1)l 2l 2(a, b)x(x 2, y 2)l 1l 1The proof of Theorem 8.3a is called a transformational proof since it usestransformations to prove the desired result.Is the converse of Theorem 8.3a also true? To demonstrate that it is, we needto show that if the slope of one line is the negative reciprocal of the slope of theother, then the lines are perpendicular. We will use an indirect proof.
The Slopes of Perpendicular Lines 309Theorem 8.3bIf the slopes of two lines are negative reciprocals, then the two lines areperpendicular.Given Two lines, AB gand AC g, that intersect atA. The slope of AB g is m and the slopeof AC gis 2m1 , the negative reciprocalof m.Prove AB g ' AC gyOBACDxProof Statements Reasons1. AC gis not perpendicular 1. Assumption.to AB g.2. Construct AD gperpendicular 2. At a point on a given line, one andto AB gat A. only one perpendicular can bedrawn.3. Slope of AB gis m. 3. Given.4. The slope of AD gis 2m1 . 4. If two lines are perpendicular, theslope of one is the negative reciprocalof the slope of the other.5. The slope of AC gis 2m1 . 5. Given.6. A, C, and D are on the same 6. Three points lie on the same line ifline, that is, AC gand AD gare and only if the slope of thethe same line.segment joining two of the pointsis equal to the slope of a segmentjoining another pair of thesepoints.7. AC g ' AB g7. Contradiction (steps 1, 6).
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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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CHAPTER1745CHAPTERTABLE OF CONTENTS
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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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Page 268 and 269: 258 Transformations and the Coordin
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Page 274 and 275: 264 Geometric InequalitiesThis corr
Page 276 and 277: 266 Geometric InequalitiesExercises
Page 278 and 279: 268 Geometric InequalitiesSubtracti
Page 280 and 281: 270 Geometric Inequalities7-3 INEQU
Page 284 and 285: 274 Geometric InequalitiesEXAMPLE 2
Page 286 and 287: 276 Geometric InequalitiesHands-On
Page 288 and 289: 278 Geometric InequalitiesABMCProof
Page 292 and 293: 282 Geometric InequalitiesGivenTo p
Page 294 and 295: 284 Geometric InequalitiesCIf the m
Page 296 and 297: 286 Geometric Inequalities7.3 If th
Page 298 and 299: 288 Geometric InequalitiesCUMULATIV
Page 300 and 301: CHAPTER2908CHAPTERTABLE OF CONTENTS
Page 302: 292 Slopes and Equations of LinesAl
Page 305 and 306: Developing SkillsIn 3-11, in each c
Page 307 and 308: y 2 bx 2 a 5 m The Equation of a Li
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Page 311 and 312: zontal line, they all have the same
Page 313 and 314: Midpoint of a Line Segment 303Proof
Page 315 and 316: Midpoint of a Line Segment 305EXAMP
Page 317: The Slopes of Perpendicular Lines 3
Page 321 and 322: Developing SkillsIn 3-12: a. Find t
Page 323 and 324: Coordinate Proof 3138-5 COORDINATE
Page 325 and 326: Coordinate Proof 315ProofThis is a
Page 327 and 328: Concurrence of the Altitudes of a T
Page 333 and 334: VOCABULARY8-1 Slope • x • y8-2
Page 335 and 336: Cumulative Review 325CUMULATIVE REV
Page 337 and 338: Cumulative Review 327Part IVAnswer
Page 339 and 340: Proving Lines Parallel 3299-1 PROVI
Page 341 and 342: Proving Lines Parallel 331In the di
Page 343 and 344: Proving Lines Parallel 333The proof
Page 345 and 346: Properties of Parallel Lines 3359-2
Page 347 and 348: Properties of Parallel Lines 337The
Page 349 and 350: gEJ. Therefore, AB g CD gbecause i
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Page 357 and 358: . Find the midpoints of two sides o
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Page 363 and 364: Proving Triangles Congruent by Angl
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Page 367 and 368: 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.
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ProofTangents and Secants 561We wil
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Tangents and Secants 563The proofs
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Developing SkillsIn 3 and 4, ABC is
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Hands-On ActivityConsider any regul
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Angles Formed by Tangents, Chords,
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Measures of Tangent Segments, Chord
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Circles in the Coordinate Plane 581
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T 3, 22lation :Circles in the Coord
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Developing SkillsIn 3-8, write an e
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Circles in the Coordinate Plane 587
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How to Proceed(1) Solve the pair of
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Tangents and Secants in the Coordin
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Chapter Summary 59321. a. Write an
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Chapter Summary 59513.12 The measur
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Vocabulary 597Formulas(Continued)Ty
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Review Exercises 59911. Two circles
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Cumulative Review 6012. The coordin
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Find the coordinatesR 90+ r y5x.Cum
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Constructing Parallel Lines 60514-1
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Constructing Parallel Lines 607Exer
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PL h b. Choose point N on . Constru
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The Meaning of Locus 611Note that i
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Five Fundamental Loci 613Applying S
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SolutionAnswerBCAB g CD g . The lo
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Points at a Fixed Distance in Coord
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Equidistant Lines in Coordinate Geo
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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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