Solution Manual For Elementary Geometry for College Students, 7th Edition

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Complete Solutions ManualPrepared byGeralyn M. KoeberleinMahomet-Seymour High School, Mathematics Department Chair, RetiredDaniel AlexanderParkland College, Professor EmeritusElementary Geometry forCollege StudentsSEVENTH EDITIONDaniel AlexanderParkland College, Professor EmeritusGeralyn M. KoeberleinMahomet-Seymour High School, Mathematics Department Chair, Retired

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iiiContentsSuggestions for Course DesignivChapter-by-Chapter CommentaryvSolutionsChapter PPreliminary Concepts1Chapter 1Line and Angle Relationships5Chapter 2Parallel Lines24Chapter 3Triangles51Chapter 4Quadrilaterals75Chapter 5Similar Triangles103Chapter 6Circles137Chapter 7Locus and Concurrence160Chapter 8Areas of Polygons and Circles177Chapter 9Surfaces and Solids210Chapter 10Analytic Geometry233Chapter 11Introduction to Trigonometry276Appendix AAlgebra Review297

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iiiSuggestions for Course DesignThe authors believe that this textbook would be appropriate for a 3-hour, 4-hour, or 5-hourcourse. Some instructors may choose to include all or part of Appendix A (Algebra Review)due to their students’ background in algebra. There may also be a desire to include TheIntroduction to Logic, found at our website, as a portion of the course requirement. Inclusionof some laboratory work with a geometry package such asGeometry Sketchpadis an optionfor course work.3-hour courseInclude most of Chapters 1–8. Optional sections could include:Section 2.2Indirect ProofSection 2.3Proving Lines ParallelSection 2.6Symmetry and TransformationsSection 3.5Inequalities in a TriangleSection 6.4Some Constructions and Inequalities for the CircleSection 8.5More Area Relationships in the Circle4-hour courseInclude most of Chapters 1–8 and include all/part of at least one ofthese chapters:Chapter 9Surfaces and Solids (Solid Geometry)Chapter 10Analytic Geometry (Coordinate Geometry)Chapter 11Introduction to Trigonometry5-hour courseInclude most of Chapters 1–11 as well as topics desired from Appendix A and/orThe Introduction to Logic (see website).Daniel C. AlexanderGeralyn M. Koeberlein

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vChapter-by-Chapter Commentary for InstructorsChapterP:Preliminary ConceptsSectionP.1:SetsandSetsofNumbersIn this section, students review the notions and basic terms related to sets of objects.Given a set or description of a set, the student should be able to classify that set as empty,finite, or infinite. For the path provided by a set of points, the student should be able tocharacterize the path as continuous or discontinuous and also to describe that path asstraight, curved, circular, or scattered. The student should recognize certain subsets of astraight line as a line segment or ray. Given two sets, the student should be able to formtheir union or their intersection. In turn, students should utilize Venn diagrams to displaytwo sets that are disjoint or the union or intersection of the two sets.SectionP.2:StatementsandReasoningThe student should realize that statements of geometry appear in both words or symbolsand can be classified as true or false. Of the compound statements (conjunction,disjunction, and implication), the instructor should warn the student of the significance ofthe implication in that it (the “If . . ., then . . .” statement) is most relevant in deductivereasoning. For the implication (also known as a conditional statement), the student shouldbe able to determine its hypothesis and conclusion; this determination acts as animportant prerequisite for preparing a proof. Also, the student should be able to recognizeand distinguish the three type of reasoning (intuition, induction, and deduction). Further,the Law of Detachment plays a major role in the development/advancement of geometry.Emphasizing that valid arguments can be confused with invalid arguments will alertstudents to potential pitfalls.Section P.3: Informal Geometry and MeasurementIn this section, many terms of geometry are introducedinformally; in Chapter 1, thesevocabulary terms will be presentedformally. For students who seem to be poorlyprepared, this approach (both an informal and a formal introduction to geometricterminology) may prove quite helpful. Measuring the line segment’s length with a rulerprepares the student intuitively for the Ruler Postulate and the Segment AdditionPostulate of Chapter 1; similarly, measuring angles with a protractor also prepares thestudent with the insights needed to deal with topics found in Section 1.2. Students thathave difficulty with measures of angles (likely due to the dual scales found onprotractors) can correct this situation by considering an activity sheet which focuses uponmeasuring angles with a protractor.Chapter One: Line and Angle RelationshipsSection 1.1: Early Definitions and PostulatesSo that the student can understand the concept “branch of mathematics,” he or she shouldbe introduced to the four parts of a mathematical system. The basic terminology andsymbolism for lines (and their subsets) must be given due attention because these will be

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viutilized throughout the textbook. The instructor should alert students to undefined termssuch as “building blocks.” Also, characterize definitions and postulates as significant inthat they lead to conclusions known as theorems, or statements that can be proven. Forthe instructor, pens and pencils can be used to visualize relationships among lines, linesegments, and rays. Table tops and pieces of cardboard can be used to represent planes.Section 1.2: Angles and Their RelationshipsIt is most important, once again, that students be able to not only recognize the terminologyfor angles, but also to be able to state definitions and principles in their own terms.Measuring angles with the protractor should enable the student to understand principlessuch as the Protractor Postulate and the Angle Addition Postulate. Constructions may alsoprovide understanding of certain concepts (like congruence and angle bisector). Manyexamples will remind the student of algebra’s role in the solution of problems of geometry.Students can be referred to the Algebra Review (Appendix A) as needed.Section 1.3: Introduction to Geometric ProofThe purpose of this section is to introduce the student to geometric proof. Many of thelittle things (hypothesis = given information, order, statements and reasons, etc.) are oftremendous importance as you prepare the student for proof. In the Sixth Edition, manyof the techniques are emphasized in the featureStrategy for Proof; be sure that yourstudents are aware of this feature and utilize these techniques. The two-column proof isused at this time because it emphasizes all the written elements of proof.Section 1.4: Relationships: Perpendicular LinesThe “perpendicular relationship” is most important to many later discoveries. For now, besure that students know that this relation extends itself to combinations such as line-line,line-plane, and plane-plane. For the general concept ofrelation, we explore the reflexive,symmetric, and transitive properties––particularly those that relate geometric figures.Some discussion ofuniquenessis productive in that it will provide background for thenotion of auxiliary lines (introduced in a later section).Section 1.5: The Formal Proof of a TheoremBe sure that your students knowin orderthe five written parts of thewritten proof:Statement of proof (the theorem), drawing (from hypothesis), given (from hypothesis),prove (from conclusion), and proof. The instructor must help the student understand thatthe unwritten Plan for Proof is far and away the most important step; for this part, suggestscratch paper, reviewing the textbook, and use of theStrategy for Prooffeature. Severaltheorems that have already been stated or proven in part are left as exercises; many ofthese have a similar counterpart (an example) in the textbook.Chapter Two: Parallel LinesSection 2.1: The Parallel Postulate and Special AnglesFrom the outset of Chapter 2, the instructor should emphasize that parallel lines must becoplanar. It is suggested that the instructor illustrate parallel and perpendicular (evenskew lines) relationships by using pens and pencils for lines and pieces of construction

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viipaper or cardboard for planes. Even though it is nearly impossible for students to graspthe significance of this fact, tell students that the Parallel Postulate characterizes thebranch of mathematics known as Euclidean Geometry (plane geometry). While thischaracterization suggests that “the Earth is flat,” it is adequate for our study even thoughspherical geometry is required at the global level. Beginning with Postulate 11, studentsshould be able to complete several statements of the form, “If two parallel lines are cut bya transversal, then . . . .”Section 2.2: Indirect ProofNote: If there is insufficient time allowed for the complete development of geometryfrom a theoretical perspective, this section can be treated as optional. This sectionprovides the opportunity to review the negation of a statement as well as the implicationand its related statements (converse, inverse, and contrapositive). Based upon thedeductive form Law of Negative Inference, the primary goal of this section is theintroduction of the indirect proof. It is important that students be aware that the indirectproof is often used in proving negations and uniqueness theorems. In the construction ofan indirect proof, the student often makes the mistake of assuming that the negation ofthe hypothesis (rather than negation of conclusion) is true.Section 2.3: Proving Lines ParallelDue to the similarity among statements of this section and those in Section 2.1, cautionstudents that parallel lines were agivenin Section 2.1. However, theorems in Section 2.3provethat lines meeting specified conditions are parallel; that is, statements in thissection take the form, “If . . . , then these lines are parallel.” For this section, havestudents draw up a list of conditions thatlead toparallel lines.Section 2.4: The Angles of a TriangleStudents will need to become familiar with much of the terminology of triangles (sides,angles, vertices, etc.). Also, students should classify triangles by using both siderelationships (scalene, isosceles, etc.) and angle relationships (obtuse, right, etc.). Somepersuasion may be needed to have students accept the use of an auxiliary line. For anauxiliary line, you must (1) explain its uniqueness, (2) verify its existence in a proof, and(3) explainwhythat particular line was chosen. The instructor cannot emphasize enoughthe role of the theorem, “The sum of the measures of the interior angles of a triangle is180°.” Because of the relation of remaining theorems to Theorem 2.4.1, notethat each statement is called acorollaryof that theorem.Section 2.5: Convex PolygonsAgain, terminology for the polygon must be given due attention. The student should beable to classify several polygons due to the number of sides (triangle, quadrilateral,pentagon, etc.). Terms such as equilateral, equiangular, and regular should be known.Rather than count the number of diagonalsDfor a polygon ofnsides, the student shouldbe able to use the formulaD=(3)2n n−. The student should be able to state and useformulas for the sum of the interior angles (or exterior angles) of a polygon; in turn, the

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viiistudent should know and be able to apply formulas that lead to the measures of an interiorangle or exterior angle of a regular polygon. Polygrams can be treated as optional.Section 2.6: Symmetry and TransformationsAppealing to the student’s intuitive sense of symmetry, the student can be taught that linesymmetry exists when one-half of the figure is the mirror image (reflection) of the otherhalf, with the line of symmetry as the mirror. For point symmetry, ask the student “isthere a point (not necessarily on the figure) that is the midpoint of a line segmentdetermined by two corresponding points on the figure in question.” While a figure mayhave more than one line of symmetry, emphasize that the figure can have only one pointof symmetry. Transformations (slides, reflections, and rotations) always produce animage (figure) that is congruent to the original figure. In Chapter 3, many examples ofpairs of congruent triangles can be interpreted as the result of a slide, reflection, orrotation of one triangle to produce another triangle (its image).Chapter Three: TrianglesSection 3.1: Congruent TrianglesAs you begin the study of congruent triangles, stress the need to pair correspondingvertices, corresponding sides, and corresponding angles. Also, students should realizethat the methods for proving triangles congruent (SSS, SAS, ASA, and AAS) are usefulthroughout the remainder of their study of geometry. Due to the simplicity and brevity ofsome proof problems found in this section, encourage students to attempt proof withoutfear. Also, have students utilize suggestions found in theStrategy for Prooffeature.Section 3.2: Corresponding Parts of Congruent TrianglesStudents should know the acronym CPCTC and know that it represents, “CorrespondingParts of Congruent Triangles are Congruent.” Emphasize that CPCTC allows them toprove that a pair of line segments (or a pair of angles) are congruent; however, warn themthat CPCTC cannot be cited as a reason unless a pair of congruent triangles have alreadybeen established. Let students know that CPCTC empowers them to take an additionalstep; for instance, proving that 2 line segments are congruent may enable the student toestablish a midpoint relationship. Once terminology for the right triangle has beenintroduced, caution students that the HL method for proving triangles congruent is validonly for right triangles. In order to give it due attention, the Pythagorean Theorem isintroduced here without proof. The connection of the Pythagorean Theorem to thissection lies in the fact that it will later be used to prove the HL theorem.Section 3.3: Isosceles TrianglesStudents should become familiar with terms (base, legs, etc.) that characterize theisosceles triangle. Students should know meanings of (and be able to differentiatebetween) these figures related to a triangle: an angle-bisector, the perpendicular-bisectorof a side, an altitude, and a median. Of course, every triangle will have three angle-bisectors, three altitudes, etc. With unsuspecting students, it may be best to show themthat the three-perpendicular bisectors of sides (or three altitudes) can intersect at a pointoutsidethe triangle; perhaps a drawing session would help! The most important theoremsof this section are converses: (1) If two sides of a triangle are congruent, then the angles

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ixopposite these sides are congruent, and (2) If two angles of a triangle are congruent, thenthe sides opposite these angles are congruent.Section 3.4: Basic Constructions JustifiedNote: If there is insufficient time or constructions are not to be emphasized, this sectioncan be treated as optional.The first goal of this section is to validate (prove) the construction methods introducedin earlier sections. For instance, we validate the method for bisecting an angle through theuse of congruent triangles and CPCTC. The second goal of this section is that ofconstructing line segments of a particular length or of constructing angles of a particularmeasure (such as 45° or 60°).Section 3.5: Inequalities in a TriangleNote: If there is insufficient time or inequality relationships are not to be emphasized, thissection can be treated as optional.To enable the proofs of theorems in this section, we must begin with a concretedefinition of the termgreater than.Note that some theorems involving inequalities arereferred to aslemmasbecause these theorems help us to prove other theorems. Theinequality theorems involving the lengths of sides and measures of angles of a triangleare very important because they will be applied in Chapters 4 and 6. For some students,the Triangle Inequality will later be applied in the coursework of trigonometry andcalculus.Chapter Four: QuadrilateralsSection 4.1: Properties of a ParallelogramAlert students to the fact that principles of parallel lines, perpendicular lines, andcongruent triangles are extremely helpful in developments of this chapter. Be sure todefine the parallelogram, but caution students not to confuse this definition with any ofseveral properties of parallelograms found in theorems of this section. These theoremshave the form, “If a quadrilateral is a parallelogram, then . . . .” In Section 4.3, theseproperties will also characterize the rectangle, square, and rhombus, because each isactually a special type of parallelogram. The final topic (bearing of airplane or ship) canbe treated as optional.Section 4.2: The Parallelogram and KiteIn this section, parallelograms arenota given in the theorems of the form, “If aquadrilateral . . . , then the quadrilateral is a parallelogram.” That is, we will be provingthat certain quadrilaterals are parallelograms. Like the parallelogram, a kite has two pairsof congruent sides; by definition, the congruent pairs of sides in the kite are adjacentsides. A kite has its own properties (like perpendicular diagonals) as well.Section 4.3: The Rectangle, Square, and RhombusConsider carefully the definition of each figure (rectangle, square, and rhombus); witheach being a type of parallelogram, the properties of parallelograms are also those of therectangle, square, and rhombus. Of course, each type of parallelogram found in this

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xsection has its own properties. For example, the rectangle and square have four rightangles while the diagonals of a rhombus are perpendicular; as a consequence of theseproperties, the Pythagorean Theorem can be applied toward solving many problemsinvolving these special types of parallelograms.Section 4.4: The TrapezoidBecause the trapezoid has only two sides that are parallel, it does not assume theproperties of parallelograms. If the trapezoid is isosceles, then it will have specialproperties such as congruent diagonals and congruent base angles. Remaining theoremsdescribe the length of a median of a trapezoid and characterize certain quadrilaterals astrapezoids or isosceles trapezoids.Quadrilateral types can be compared by use of a Venn diagram or the following outline:1.QuadrilateralsA.Parallelograms1.Rectanglea.Square2.RhombusB.KitesC.Trapezoids1.Isosceles TrapezoidsChapter Five: Similar TrianglesSection 5.1: Ratios, Rates, and ProportionsNote: For work in Chapter 5, the instructor may want to refer those students who need areview of the methods of solving quadratic equations to Appendix Sections A.4 and A.5.In this section, emphasize the difference between a ratio (quotient comparinglikeunits) and a rate (quotient comparingunlike units).Students should understand that aproportion is an equation in which two ratios (or rates) are equal. Terminology forproportions (means, extremes, etc.) are important because the student better understandsa property like the Means-Extremes Property.Section 5.2: Similar PolygonsIn this section, similar polygons are defined and their related terminology (correspondingsides, corresponding angles, etc.) are introduced. The definition of similar polygonsallows students to (1) equate measures of corresponding angles, and (2) form proportionsthat compare lengths of corresponding sides. Thus, this section focuses on problemsolving strategies, including an ancient technique known asshadow reckoning.Section 5.3: Proving Triangles SimilarWhereas Section 5.2 focuses on problem solving, Section 5.3 emphasizes methods forproving that triangles are similar. Due to its simplicity, the instructor should emphasizethat the AA method for proving triangles similar should be used whenever possible. Thedefinition of similar triangles forces two relationships among parts of similar triangles:(1.) CASTC means “Corresponding Angles of Similar Triangles are Congruent,” while(2.) CSSTP means “Corresponding Sides of Similar Triangles are Proportional.”

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xiOther methods for proving triangles similar are SAS and SSS; in application, thesemethods are difficult due to the necessity of showing lengths of sides to be proportional.Warn students not to use SAS and SSS (methods of proving triangles congruent) asreasons for claiming that triangles are similar.Section 5.4: The Pythagorean TheoremTheorem 5.3.1 leads to a proof of the Pythagorean Theorem and its converse. Studentsshould be aware that many (more than 100) proofs exist for the Pythagorean Theorem.For emphasis, note that the Pythagorean Theorem allows one to find the length of a sideof a right triangle; however, its converse enables one to conclude that a given trianglemay be a right triangle. Because these are commonly applied, Pythagorean Triples suchas (3,4,5) and (5,12,13) are best memorized by the student.Withcbeing the length of the longest side of a given triangle, this triangle is:anacutetriangle ifc2<a2+b2, arighttriangle ifc2=a2+b2,or anobtusetriangle ifc2>a2+b2.Section 5.5: Special Right TrianglesIn Section 5.4, some right triangles were special because of their integral lengths of sides(a,b,c). In Section 5.5, a right triangle with angle measures of 45°, 45°, and 90° alwayshas congruent legs while the hypotenuse is2times as long as either leg. Also, a righttriangle with angle measures of 30°, 60°, and 90° has a longer leg that is3 times as longas the shorter leg, while the hypotenuse is two times as long as the shorter leg. Theserelationships, and their converses, also have applications in trigonometry and calculus.Section 5.6: Segments Divided ProportionallyNote: In this section, Ceva’s Theorem is optional in that it is not applied in later sections.The phrasedivided proportionallycan be compared to profit sharing among unequalpartners in a business venture. This concept is, of course, the essence of numerousapplications found in this section. The Angle-Bisector Theorem states that an angle-bisector of an angle in a triangle leads to equal ratios among the parts of the lengths of thetwo sides forming the bisected angle and the lengths of parts of the third side.Chapter Six: CirclesSection 6.1: Circles and Related Segments and AnglesTerminology for the circle is reviewed and extended in this section. Students will havedifficulty with the definition ofcongruent arcsin that they must have both equalmeasures and lie within thesamecircle orcongruentcircles. Many of the principles ofthis section are intuitive and therefore easily accepted. Contrast the sides and vertexlocations of the central angle and the inscribed angle. Stress these angle-measurementrelationships in that further angle-measurement relationships will be added in Section 6.2.Section 6.2: More Angle Measures in the CircleThe termstangentandsecantare introduced and will be given further attention in latersections as well as in the coursework of trigonometry and calculus. Again emphasize the

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xiinew angle-measurement techniques with the circle. A summary of methods (Table 6.1) isprovided for students.Section 6.3: Line and Segment Relationships in the CircleThe early theorems in this section sound similar, yet make different assertions; for thisreason, it may be best that students draw the hypothesis of each theorem to “see” that theconclusion must follow. Students will also need to distinguish the concepts ofcommontangent for two circlesandtangent circles.Each of the relationships found in Theorems6.3.5–6.3.7 is difficult to believe without proof; however, with the help of an auxiliaryline, each proof of theorem is easily and quickly proved.Section 6.4: Some Constructions and Inequalities for the CircleNote: If there is insufficient time or constructions and inequality relationships are not tobe emphasized, this section can be treated as optional.Because the construction methods of this section are fairly involved, be sure to assignhomework exercises that have students perform them. The inequality relationshipsinvolving circles are intuitive (easily believed); due to the difficulty found in constructingproofs of these theorems, the instructor may wish to treat proofs as optional.Chapter Seven: Locus and ConcurrenceSection 7.1: Locus of PointsSo that the termlocusis less confusing for students, the instructor should tie this word toits Latin meaning: “location.” For the locus concept, quantity makes a difference; that is,students will need to see several examples. While construction of a locus is optional, adrawing of the locus is imperative. Theorems 7.1.1 and 7.1.2 are most important in thatthey lay the groundwork for later sections. The instructor should be sure to distinguishbetween the locus of points in a plane and the locus of points in space.Section 7.2: Concurrence of LinesThe discussion of locus leads indirectly to the notion of concurrence. In particular, theconcurrence of the three angle-bisectors of a triangle follows directly from the first locustheorem in Section 7.1; in turn, a triangle has an inscribed circle whose center is theincenter of the triangle. Likewise, the three perpendicular-bisectors of the sides of atriangle are concurrent at the circumcenter of the triangle, the point that is the center ofthe circumscribed circle of every triangle. In this section, not only have studentsmemorize the termsincenter,circumcenter,orthocenter, andcentroid, but also have themknow which concurrency (angle-bisectors, etc.) leads to each result.Section 7.3: More About Regular PolygonsBased upon our findings in Section 7.2, the student should know that a circle can beinscribed ineverytriangle and also be circumscribed abouteverytriangle. Further, thecenter for both circles (inscribed and circumscribed) is the same point for the equilateraltriangle and regular polygons in general. The new terminology for the regular polygon(center, radius, apothem, central angle, etc) should be memorized because it will also beapplied in Section 8.3.

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xiiiChapter Eight: Areas of Polygons and CirclesSection 8.1: Area and Initial PostulatesEven though most students sayarea of a triangle,they should realize that the moreaccurate description would bearea of triangular region.Stress the difference betweenlinear units (used to measure length) and square units (used to measure area). With eacharea formula serving as a “stepping stone” to the next formula, the given order for thearea formulas is natural. Perhaps the most significant formula in the list is that of theparallelogram (A=bh) in that this is derived from the area of rectangle formula while itleads to the remaining formulas.Section 8.2: Perimeter and Area of PolygonsGiven its practical applications, the notion of perimeter should be reviewed and extended.Heron’s Formula is difficult to state and apply; however, it is common to find the area ofa triangle whose lengths of sides are known. The proof of Heron’s Formula is found atthe website that accompanies this textbook. Emphasize Theorem 8.2.3 and that the areaformulas for the rhombus and kite are just special cases of this theorem.Section 8.3: Regular Polygons and AreaIn this section, we first consider formulas for the area of the equilateral triangle andsquare. For regular polygons in general, be sure to introduce or review the terminology(center, radius, apothem, central angle, etc.) that was found in Section 7.3; if studied, thework in both Chapters 9 and 11 use this terminology as well. The ultimate goal of thissection is to establish the formula for the area of a regular polygon, namelyA= 12aP.Section 8.4: Circumference and Area of a CircleBegin with the definition ofπas a ratio and then provide some approximations of itsvalue (such as227and 3.1416). Usingπ=Cd, we can show thatC=πdandC= 2πr.Using a proportion, we find the length of an arc of circle (as part of the circumference).Developed as the limit of areas of inscribed regular polygons, we show that the area of acircle is given byA=πr2. Note that the concept oflimitneeds a few examples. Forstudents to distinguish between 2πrandπr2(for circumference and area), have themcompare units, wherer= 3 cm, 2πr= 2π⋅3 cm = 6πcm (a linear measure) whileπr2=π⋅3 cm⋅3 cm or 9πcm2(a measure of area).Section 8.5: More Area Relationships in the CircleNote: If there is insufficient time for the study of this section, it can be treated as optionalin that none of the content is used in later sections.Formulas for the area of a sector and segment depend upon the formula for the area ofa circle; however, an understanding of these area concepts is far more important than thememorization of formulas. The area of segment applications require that the relatedcentral angle have a convenient measure, such as 60°, 90°, or 120°; otherwise,

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xivtrigonometry would be necessary to solve the problem. When a triangle has perimeterPand inscribed circle of radiusr, the area of the triangle is given byA= 12rP.Chapter Nine: Surfaces and SolidsSection 9.1: Prisms, Area, and VolumeThe student should consider three-dimensional objects in this section and chapter; for thatpurpose, the instructor should use a set of models displaying various prisms and othersolids or space figures. Students need to become familiar with prisms and relatedterminology. To calculate the lateral area and the total area of a prism, a student mustapply formulas from Chapter 8. For the volume formula for a prism (V=Bh), emphasizethatBis the area of the base and thatVis always measured incubic units.Section 9.2: Pyramids, Area, and VolumeAgain, the instructor should use a set of models to display various pyramids. Studentsneed to become familiar with pyramids and related terminology, including theslantheightof a regular pyramid. Calculating the lateral area and the total area of a pyramidrequires the application of formulas from Chapter 8. To find the length of the slant heightof a regular pyramid requires the use of the Pythagorean Theorem. Compare the formulafor the volume of a pyramid (V= 13Bh) to that of the prism (V=Bh).Section 9.3: Cylinders and ConesComparing the prism to cylinder and the pyramid to cone will help to motivate studentsin learning the area and volume formulas of this section. Three-dimensional models willmotivate the formula for the lateral area of cylinder and to explain the slant height of theright circular cone. The length of the slant height of the right circular cone can be foundby using the Pythagorean Theorem. Compare volume formulas for the prism (V = Bh)and right circular cylinder (V=BhorV=πr2h); likewise compare the volume formulasfor the pyramid (V= 13Bh) and right circular cylinder (V= 13BhorV= 13πr2h). Whilethe material involving solids of revolution is a preparatory topic for calculus, it can betreated as optional.Section 9.4: Polyhedrons and SpheresStudents should recognize (or be told) that prisms and pyramids are merely examples ofpolyhedrons (or polyhedra). Students should verify Euler’s Formula (V + F = E + 2) forpolyhedra with a small number of vertices by using solid models from a kit. For thesphere, compare its terminology with that of the circle; however, note that a sphere alsohas tangent planes. To develop the volume of sphere formula, it is necessary to interpretthe volume as the limit of the volumes of inscribed regular polyhedra with anincreasing number of faces. Due to limitations, we only apply the surface area of sphereformula.

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xvChapter Ten: Analytic GeometrySection 10.1: The Rectangular Coordinate SystemThe student should become familiar with the rectangular coordinate system and its relatedterminology. Warn students that the definitions for lengths of horizontal and vertical linesegments are fairly important in the development of the chapter. For the formulas developed,P1is readfirstpoint and x1as thevalue of x for the first point.The Distance Formula andMidpoint Formulas are must be memorized in that they will be used throughout Chapter 10;of course, these formulas are also useful in later coursework as well.Section 10.2: Graphs of Linear Equations and SlopeAt first, a point-plot approach for graphing equations is used. However, graphing linearequations leads to graphs that are lines and, in turn, the notion ofslopeof a line. Thestudent must memorize the Slope Formula. By sight, a student should be able torecognize that a given line has a positive, negative, zero, or undefined slope.Many students have difficulty drawing a line based upon its provided slope; for thisreason, it is important to treat slope as m =riserun. To draw a line with slope m, move fromone point to the second point by simultaneously using a vertical change (rise) thatcorresponds to the horizontal change (run). Using the slopes of two given lines, thestudent should be able to classify lines as parallel, perpendicular, or neither.Section 10.3: Preparing to Do Analytic ProofsThis section is a “warm up” for completing analytic proofs that follow in Section 10.4.Specific goals that need to be achieved are:1.The student should know the formulas found in the summary on the first page.2.The student should follow the suggestions for placement of a drawing so that theproof of the theorem can be completed. See theStrategy for Proof.3.The student should study the relationship between desired theorem conclusions andformulas needed to obtain such conclusions. See theStrategy for Proof.Section 10.4: Analytic ProofsThis section utilizes all formulas and suggestions from previous sections of Chapter 10.In each classroom, the instructor must warn students of the amount of rigor required. Forinstance, suppose that we are trying to prove a theorem such as, “If a quadrilateral is aparallelogram, then its diagonals bisect each other.” Does the student provide a figurewith certain vertices that is known to be a parallelogram,ordoes that figure have to beproven a parallelogram before the proof can be continued? You may wish to prove eachclaim once and then accept it at a later time as given (not needing proof); if it was shownin an earlier section that the triangle with vertices atA(−a,0),B(a,0), andC(0,b) isisosceles, then it will be given as such in a later proof.Section 10.5: Equations of LinesIn this section, we use given information about a line (like slope andy-intercept) to findits equation. Students will need to memorize and apply both the Slope-Intercept and thePoint-Slope forms of a line. Emphasize that solving systems of linear equations is the

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xvialgebraic equivalent of finding the point of intersection of two lines using geometry.Emphasize that the method (algebra or geometry) used to find this point of intersectionalways leads to the same result. Point out that the Slope-Intercept and Point-Slope formsof a line can be used to prove further theorems by the analytic approach.Section 10.6: The Three-Dimensional Coordinate SystemIn this section, we plot points of the form (x,y,z) in three dimensions. Warn students thatthe forms of equations of a line will seem unfamiliar; however, the equation of a plane inCartesian space is similar to the general form for the equation of line in the Cartesianplane. Students should easily adapt to the natural extensions of the Distance Formula andMidpoint Formula. Ironically, there is no Slope Formula. For the concept of directionvector, the student will need some convincing of its importance; however, the directionvectors for two lines will determine whether these lines have the same direction (parallelor coincident) or different directions (intersecting or skew). To consider the relationshipsbetween planes, the instructor will need to give considerable attention to algebraictechniques in that the methods will be more involved. This section concludes with theequation of a sphere in Cartesian space, again seen by students as a natural extension ofthe equation of a circle in the Cartesian plane.Chapter Eleven: Introduction to TrigonometrySection 11.1: The Sine Ratio and ApplicationsRelated to the right triangle, ask students to memorize the sine ratio of an angle in theformoppositehypotenuse; while this seems rather informal, the remaining definitions oftrigonometric ratios will be given in a similar form. While students are encouraged to usethe calculator to find sine ratios for angles, they should also know these results frommemory: sin 0° = 0, sin 30° = 12 , sin 45° =22, sin 60° =32, and sin 90° = 1.Students should realize that the sine ratios increase as the angle measure increases.Emphasize the termsangle of elevationandangle of depressionand be able to performapplications that require the use of the sine ratio.Section 11.2: The Cosine Ratio and ApplicationsAsk students to memorize the cosine ratio of an angle in the formadjacenthypotenuse.In addition to using the calculator to find cosine ratios, students should memorize resultssuch as cos 0° = 1, cos 30° =32, etc. Students should recognize that an increase inangle measures produces a decrease in cosine measures. Students need to be able tocomplete applications that require the cosine ratio. The instructor should include andperhaps require that the student be able to prove the theorem sin2θ+ cos2θ= 1.Emphasize that many geometry problems (such as Example 7 of this section) cannot besolved without the use of trigonometry.

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