CP1Math2 Name_____________________________

CP1Math2MidtermExamReviewDeductiveGeometry(Ch.6)• Writinggeometricproofs• Trianglecongruenceanditsuses• Parallellinesandtheiruses• QuadrilateralsandtheirpropertiesExponents&Radicals(Ch.1)ArithmeticwithRadicals

• Adding,subtracting,multiplying,dividing–includesrationalizingthedenominator• Simplifyingradicals• Solvesimpleequations( 3 + 48 = k 3 )• Classifynumbers(real,rational,irrational,integer)

PropertiesofExponents(useinbothdirections)Solvingexponentialequations(changetosamebaseandsolve)Quadratics&Polynomials(Ch.2&3)SolvingQuadraticEquations3methodstosolvequadraticsequations:

• Factoringo Typesofquadraticswefactored:differenceof2perfectsquares,monicsandnon-monics(with

andwithoutGCF’s)o Methods:factoroutGCF;sumsandproducts(monic),splittingthemiddleterm,z-substitution

• Completingthesquare• Quadraticformula

Connectionsbetweenequationsandgraphsofquadratics

• x-intercepts,roots,zerosaresolutionstothequadraticwhenequalto0• y-interceptàpointony-axis(i.e.f(0))• lineofsymmetryàaverageoftheroots• vertexàmaximumorminimumofparabola,(avgroots,f(avgroots))orcompletethesquare• otherpointsàusesymmetryofgraphtofindadditionalpoint(s)

Formatsforquadraticfunctions&abilitytomovebetweenforms:

• StandardForm:𝑦 = 𝑎𝑥! + 𝑏𝑥 + 𝑐• FactoredForm:𝑦 = 𝑎(𝑥 − 𝑟)(𝑥 − 𝑠)• VertexForm:𝑦 = 𝑎(𝑥 − ℎ)! + 𝑘

o Standardformàvertexform(completesquare)o Standardformàfactoredform(factor)o Factoredorvertexformàstandardform(multiplyandsimplify)

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PracticeProblems: (Completeonaseparatesheetofpaper)1.Given:𝐵𝐶𝐷𝐸 isanisoscelestrapezoidwithlegs𝐷𝐸and𝐶𝐵.Prove:∠1 ≅ ∠22. Given:∠3 ≅ ∠4;∠1 ≅ ∠2 Prove:𝐴𝑋 ≅ 𝐵𝑌

3. Given: 𝐴𝐹 ≅ 𝐴𝐸 𝐹𝐷 ≅ 𝐸𝐷 ∠1 ≅ ∠2 Prove: 𝐴𝐷 ⊥ 𝐶𝐵4. Given:Parallelogram𝐴𝐵𝐶𝐷 withdiagonalsintersectingatpoint𝑂. Prove:𝑋𝑂 ≅ 𝑌𝑂

5.RondaandStephanieareworkingtoprovethat𝐴𝐶 isparallelto𝐵𝐷 in

thediagramtotheright.Ronda’sargumentis:

𝐴𝐶 ∥ 𝐵𝐷 because:∡ 1 = ∡2+ ∡3and∡1 = ∡4+ ∡5 bytheexterioranglesumtheorem.Therefore∡3 = ∡4.SincethesearealternateinterioranglesandtheyarecongruentweknowbyAIPthatthelinesareparallel.

WhereistheincorrectstepinRonda’sproof?

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6.Given:𝐴𝐵 ≅ 𝐵𝐶 𝐴𝐷 ≅ 𝐷𝐶 Prove:∠1 ≅ ∠27.Given:𝐽𝑈𝑀𝑃isaparallelogram;𝑆,𝑇,𝐴,𝑎𝑛𝑑 𝑅aremidpoints.

a.Usinggeometricrelationshipsexplainwhy𝑆𝑇𝐴𝑅 isnotnecessarilyarhombus.b.Whatmustbetrueinorderfor𝑆𝑇𝐴𝑅 tobearhombus?

8.Foreachequation,findthevalueofkthatsatisfiestheequation.

9.Simplifyasmuchaspossible.

a. 𝑥! ! !! b. 𝑥!" ∙ !!!

! c. 𝑥! ∙ 𝑥! !

d. 3𝑥!𝑦! ! e. !!!!!

!!!! f.

! !!

g. 125! h. !!!

!!! i. !!

10.SolveeachequationusingfactoringandZPP,quadraticformula,orcompletingthesquare.Trytopracticeallthemethodsacrossthe8problems.

a.3𝑥! − 48 = 0 b.𝑥! + 4𝑥 = −4 c.3𝑥! − 8 = 2𝑥 d.5𝑥! − 20𝑥 = 60 e.2𝑥! + 5𝑥 = 7f.3(𝑥 + 5)! = 9g.2𝑥! + 14 = 20𝑥 h.3𝑥! = −9𝑥 − 3

11.Hereisaquadraticinstandardform:𝑦 = 3𝑥! + 12𝑥 − 15Withoutusingagraphingcalculator,sketchagraphclearlylabeling:

• Thevertex• Thex-intercepts• Theaxisofsymmetry• They-intercept• Theotherpointonthegraphwiththesamey-valueasthey-intercept.

a. 98 = 𝑘 2

b. 3 ∙ 75 = 𝑘

c. 5+ 125 = 𝑘 5

d. 2 6+ 150 = 𝑘 6

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12.Consideranon-monicquadraticwithavertexat(3,−2)andthatalsocontainsthepoint(0,−6.5).Writeanequationinanyformtorepresentthefunctiondescribed.

13.Let𝑓 𝑥 = 𝑥! − 3𝑥! − 16𝑥. a. Whatisthedegreeof𝑓(𝑥)?

b. Find𝑔(𝑥)suchthat𝑓 𝑥 − 𝑔 𝑥 = −3𝑥! − 𝑥! + 4𝑥 − 14. c.Findh(x)suchthat𝑓 𝑥 + ℎ(𝑥)isafourthdegreepolynomialwithalineartermof9. d.Findj(x)suchthat𝑓 𝑥 ∙ 𝑗(𝑥)isafifthdegreepolynomialwithaleadingcoefficientof6.

e.Findthezerosof𝑔(𝑥).Hint:yourworkon#10gishelpfulheref.Useyourzerosfrompartctofindthevertexof𝑔(𝑥).

14.Aballisthrownupwardfromaflatsurface.Itsheightinfeetasafunctionoftimesinceitwasthrown(in

seconds)isgivenbytheequationℎ(𝑡) = −16𝑡! + 34𝑡 + 25.Youmaycalculatorfeaturestoanswerthequestionsbelow.

a.Evaluateℎ(1),andexplainwhatitmeansinthecontextofthisproblem.b. Whatwasthehighesttheballgot,andwhendiditreachthatheight?c. Atwhattimedoestheballland?

15.Thediagramattherightisagraphof𝑓(𝑥) = −2𝑥! + 20𝑥 − 32.

∆𝐴𝐵𝐶 hasverticesatthexinterceptsandvertexoftheparabola.

a.Findtheareaof∆𝐴𝐵𝐶b.FindthelocationofpointDthatmakesquadrilateral𝐴𝐵𝐶𝐷arhombus.c.Findaquadraticfunctionwhosevertex,pointE,wouldmakequadrilateral𝐴𝐵𝐶𝐸akite.

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Answers:1.

Given Prove1.𝐵𝐶𝐷𝐸 isanisoscelestrapezoidwithlegs𝐷𝐸and𝐶𝐵

1.Given

2.𝐴𝐷 ∥ 𝐸𝐵 2.Definitionofatrapezoid3.∠1 ≅ ∠𝐶𝐵𝐸 3.Thebaseanglesofanisoscelestrapezoid

arecongruent4.∠𝐶𝐵𝐸 ≅ ∠2 4.PAI5.∠1 ≅ ∠2 5.Transitiveproperty2.

Given Prove1.∠3 ≅ ∠4;∠1 ≅ ∠2 1.Given2.∠𝐴𝑋𝑌 = ∠4 + ∠2 ,∠𝐵𝑌𝑋 = ∠3 + ∠1 2.Wholeisthesumofitsparts3.∠𝐴𝑋𝑌 ≅ ∠𝐵𝑌𝑋 3.AdditionProperty4.𝑋𝑌 ≅ 𝑌𝑋 4.ReflexiveProperty5.Δ𝐴𝑋𝑌 ≅ Δ𝐵𝑌𝑋 5.ASA6. 𝐴𝑋 ≅ 𝐵𝑌 6.CPCTC3.

Statement Reason1.𝐴𝐹 ≅ 𝐴𝐸,𝐹𝐷 ≅ 𝐸𝐷,∠1 ≅ ∠2 1.Given2.𝐴𝐷 ≅ 𝐴𝐷 2.ReflexiveProperty3.∆𝐸𝐴𝐷 ≅ ∆𝐹𝐴𝐷 3.SSS4.∠𝐸𝐷𝐴 ≅ ∠𝐹𝐷𝐴 4.CPCTC5.∠𝐶𝐷𝐴 = ∠1 + ∠𝐸𝐷𝐴and∠𝐵𝐷𝐴 = ∠2 + ∠𝐹𝐷𝐴 5.Whole=sumofparts6.∠𝐶𝐷𝐴 ≅ ∠𝐵𝐷𝐴 6.Additionproperty7.∠𝐶𝐷𝐴and∠𝐵𝐷𝐴aresupplementary 7.Anglesthatformastraightlineare

supplementary(definitionofsupplementary)8.∠𝐵𝐷𝐴 = ∠𝐶𝐷𝐴 = 90° 8.Supplementaryandcongruentanglesare90°9.𝐴𝐷 ⊥ 𝐶𝐵 9.Definitionofperpendicular

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Given Prove1.Parallelogram𝐴𝐵𝐶𝐷 withdiagonalsintersectingatpoint𝑂.

1.Given

2.𝐴𝑂 ≅ 𝐶𝑂 2.Thediagonalsofaparallelogrambisecteachother

3.∠𝐴𝑂𝑋 ≅ ∠𝐶𝑂𝑌 3.VAT4. 𝐴𝐵 ∥ 𝐷𝐶 4.Oppositesidesofaparallelogramare

parallel(Definitionofparallelogram)5. ∠𝑌𝐶𝑂 ≅ ∠𝑋𝐴𝑂 5.PAI6.Δ𝐴𝑂𝑋 ≅ Δ𝐶𝑂𝑌 6.ASA7.𝑋𝑂 ≅ 𝑌𝑂 7.CPCTC

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5.Thestepthatisincorrectis:Therefore∡3 = ∡4.Thoughitistruethat∡ 1 = ∡2+ ∡3and∡1 = ∡4+ ∡5,making∡4+ ∡5 = ∡2+ ∡3bythetransitiveproperty.Itisnottruethatanyoftheindividualanglesneedtobeequaltoeachother.Forexample20+70=40+50istrue.Bothsumsequal90,butnoneofthenumbersintheequationareequal.

6.Statement Reason1.𝐴𝐷 ≅ 𝐷𝐶,𝐴𝐵 ≅ 𝐵𝐶 1.Given

2.∠3 ≅ ∠4,∠𝐵𝐴𝐶 ≅ ∠𝐵𝐶𝐴 2.ITT3.∠𝐵𝐴𝐶 = ∠1 + ∠3and∠𝐵𝐶𝐴 = ∠2 + ∠4 3.Wholeisthesumofitsparts4.∠1 + ∠3 ≅ ∠2 + ∠4 4.TransitiveProperty5.∠1 ≅ ∠2 5.SubtractionProperty

7.a.Theoppositesidesandanglesareequalinaparallelogram,sohalvesofequalsideswouldalsobeequal.

ThiswouldmakeΔ𝑆𝑈𝑇 ≅ Δ𝐴𝑃𝑅bySASsothat𝑆𝑇 ≅ 𝐴𝑅byCPCTC.Also,Δ𝑆𝐽𝑅 ≅ Δ𝐴𝑀𝑇bySASsothat𝑆𝑅 ≅ 𝐴𝑇byCPCTC.Thisdoesnotmeanthat𝑆𝑅 ≅ 𝐴𝑇 ≅ 𝑆𝑇 ≅ 𝐴𝑅whichwouldhavetobetruetomaketheSTARarhombus.

b.InorderforSTARtobearhombus,∡𝑈 ≅ ∡𝑃 ≅ ∡𝐽 ≅ ∡𝑀.Inotherwords,JUMPmustbearectangle.8. a.k=7 b.k=15 c.k=6 d.k=7

9. a. 𝑥!! = !!! b. 𝑥!! = !

!! c.𝑥!" d.27𝑥!"𝑦! e.3𝑥!𝑧!

f.2𝑥!!/! = !! g.5 h.𝑥 i.! !

!

10. Suggestedmethodsgiven: a.4,-4 Solvingdirectlyistheeasiestway b.-2 Thisisaperfectsquarealready. c.!!

!,2 Splittingthemiddle

d.6,-2 FactoroutaGCFandthenusesum/product e.!!

!,1 Splittingthemiddle

f.−5± 3 Dividebothsides3,thentakethesquarerootofbothsides.g.5± 3 2 Completingthesquare

h.!𝟑± !𝟐

QuadraticFormula11.vertex:(-2,-27),x-intercepts:(-5,0)and(1,0),axisofsymmetry:x=-2,y-intercept:(0,-15), symmetricpoint:(-4,-15)youcancheckyourgraphonagraphingcalculator12.𝑦 = −!

! 𝑥 − 3! − 2

13.a. degree3 b. 𝑔 𝑥 = 2𝑥! − 20𝑥 + 14 c.needfourthdegreewith25x:onepossibility: ℎ 𝑥 = 𝑥! + 25𝑥

d.needquadraticwithleadingcoefficientof-2:onepossibility: 𝑗 𝑥 = −2𝑥!e. 𝑥 = 5± 3 2

f. (5,−36)14.a.ℎ(1) = 43,meaningtheballis43feethighafter1second b.1.063seconds,43.063feet c.2.703seconds15.a.!! 6 18 = 54units2 b.(5,–18) c.needvertexatx=5:onepossibility:y=–2(x–5)2–3