ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020

NAME:________________________________________ThispackethasbeendesignedtohelpyoureviewvariousmathematicaltopicsthatarenecessaryforyoursuccessinAlgebraIand/orGeometrynextyear,aswellasPSAT/SAT/ACT.Completeallproblemsandshowyourwork!ItwillbecollectedforagradebyyourAlgebraI/Geometryteacherthefirstweekofschool.Thereareworked-outexampleproblemswithinmostofthesections.

TOPIC1:EvaluatingExpressionsandEquations

Example1 Example2 Example3Findthevalueoftheexpression𝟒𝒙𝒚 + 𝟐𝒚𝟐when𝒙 = 𝟑𝐚𝐧𝐝𝒚 = −𝟓

4(3)(−5) + 2(−5)"=−60 + 2(25)=−60 + 50=−𝟏𝟎

Findthevalueoftheexpression!𝟒𝒙𝟐𝒙%𝟓𝒚

when𝒙 = −𝟐𝐚𝐧𝐝𝒚 = 𝟑

−4(−2)2(−2) + 5(3)

=8

−4 + 15

=𝟖𝟏𝟏

Findtheareaofacirclewhoseradiusis4inches.

𝑨 = 𝝅𝒓𝟐

𝐴 = 𝜋(4)"𝐴 = 𝜋(16) = 16𝜋𝑨 = 𝟓𝟎. 𝟐𝟕𝒊𝒏𝟐

Note:youshouldusethe𝜋buttononyourcalculatortogetthemostaccurateanswer

Evaluateeachexpressionwhen𝒙 = 𝟓𝐚𝐧𝐝𝒚 = −𝟒.Showyourwork!(SeeExamples1and2)1.−8𝑥𝑦 2.𝑥" −𝑦" 3.5𝑥" − 2𝑥𝑦

4.9𝑥"𝑦 − 4𝑦 5.()

*+%,)+ 6.

)!+-)+

Evaluatethefollowingformulasgiventhevariablevalues.(SeeExample3)7.𝑷 = 𝟐𝒘+ 𝟐𝒍;𝑤 = 15, 𝑙 = 12 8.𝑨 = 𝝅𝒓𝟐;𝑟 = 69.𝑺 = 𝟐𝝅𝒓𝒉; 𝑟 = 3, ℎ = 7 10.𝑽 = 𝝅𝒓𝟐𝒉; 𝑟 = 5, ℎ = 811.𝑽 = 𝟏

𝟑𝝅𝒓𝟐𝒉; 𝑟 = 3, ℎ = 12 12.𝑨 = 𝟏

𝟐𝒉(𝒃𝟏 +𝒃𝟐); ℎ = 4.5, 𝑏% = 7, 𝑏" = 8

ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020 TOPIC2:SolvingLinearEquations

Example1 Example2 Example3Solveforthevariable:𝟕𝒙 − 𝟗 = 𝟏𝟗+ 9 + 9 add 9 to both sides 𝟕𝒙 = 𝟐𝟖 77 divide by 7 on both sides 𝒙 = 𝟒

Solveforthevariable:𝟖𝒈 + 𝟏𝟏 = 𝟔𝒈 + 𝟑𝟏-6g -6g move lower variable 𝟐𝒈 + 𝟏𝟏 = 𝟑𝟏 -11 -11 subtract 11 from both sides 𝟐𝒈 = 𝟐𝟎 divide by 2 on both sides 𝒈 = 𝟏𝟎

Solveforthevariable:𝟐(𝟓 + 𝟑𝒙) = 𝟒(−𝟏𝟎 − 𝒙)𝟏𝟎 + 𝟔𝒙 = −𝟒𝟎 − 𝟒𝒙distribute + 4x + 4x move lower variable 𝟏𝟎 + 𝟏𝟎𝒙 = −𝟒𝟎 -10 -10 subtract 10 from both sides 𝟏𝟎𝒙 = −𝟓𝟎 divide by 5 on both sides 𝒙 = −𝟓

Solvethefollowingequationsforthevariable.1.5𝑥 + 1 = 31 2.3𝑥 − 4 = 8 3.7𝑥 = 50 + 2𝑥4.6𝑥 + 7 = 8𝑥 − 13 5.10𝑔 + 20 = 7𝑔 − 10 6.13.7𝑏 − 6.5 = −2.3𝑏 + 8.37.−7(𝑥 − 3) = −4 8.&

"𝑦 − 1 = %

"𝑦 + 8 9.6(𝑎 + 2) − 4 = −10

10.−8(4 + 9𝑥) = 7(−2 − 11𝑥) 11.4(2𝑐 − 8) = %

'(49𝑐 + 70)

12.Ifthelengthofarectangleisthreetimesitswidthanditsperimeteris24cm,whatisitsarea?

ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020 TOPIC3:PythagoreanTheorem:Ifthelengthsofthelegsofarighttriangleareaandb,andthelengthofthehypotenuse(sideoppositefromtherightangle)isc,then𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐. 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐

Note:Legsaandbareinterchangeable,butcMUSTbethehypotenuse

Example1 Example2 Example3x𝑎" + 𝑏" = 𝑐"PythagoreanTheorem7" +8" =𝑥"Substitute49 + 64 = 𝑥"Multiply113 = 𝑥"Add𝒙 = 𝟏𝟎. 𝟔𝟑Takethesquareroot

𝑎" + 𝑏" = 𝑐"PythagoreanTheorem𝑥" +9" =12"Substitute𝑥" + 81 = 144Multiply𝑥" = 63Subtract81fromboth𝒙 = 𝟕. 𝟗𝟒Takethesquareroot

A13-footladderleansagainstawall,withitsbase5feetawayfromthewall.Howhighuponthewalldoestheladderreach?𝑎" + 𝑏" = 𝑐"𝑥" +5" =13"𝑥" + 25 = 169𝑥" = 144𝒙 = 𝟏𝟐𝒇𝒆𝒆𝒕

Solveforthemissingside.Roundalldecimalstothenearesthundredth.1. 2. 3.4. 5. 6*.

ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020 For#7-8,drawadiagrammodelingthescenarioandsolve.Roundalldecimalstothenearesttenth.7.A15-footladderisleaningagainstatree.Iftheladderreaches10feethighonthetree,howfaristhebaseoftheladderfromthetree?8.Marialeavesherhouseandwalksnorthfor6blocks.Shethenturnsandwalksdirectlyeastfor8moreblocksuntilshereachesNatalie’shouse.WhatistheshortestdistancebetweenMaria’shouseandNatalie’shouse?9.Askateboardrampcoversahorizontaldistanceof17feetandraisesto4feetabovetheground.Whatisthelengthoftheskateboardramp?TOPIC4:DistanceFormula:ThedistanceformulaisderivedfromthePythagoreanTheorem.Tofindthedistancebetweentwopoints(𝑥%, 𝑦%)𝑎𝑛𝑑(𝑥", 𝑦"),usethecoordinatesoftheorderedpairsandapplytheformula:

𝒅 = -(𝒙𝟐 −𝒙𝟏)𝟐 + (𝒚𝟐 −𝒚𝟏)𝟐

Example1 Example2 Determineifthetrianglebelowisisosceles(twosideswiththesame

length).Thistriangleisnotisoscelessinceallthreesidesaredifferentlengths

Findthedistancebetweenthegivenpoints.Roundalldecimalstothenearesthundredth.1.(4, 8)𝑎𝑛𝑑(6, 12) 2.(10, 7)𝑎𝑛𝑑(−3, 1) 3.(3, −8)𝑎𝑛𝑑(−5, 0)

ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020 4.(4, −5)𝑎𝑛𝑑(−6,−1) 5.(2, −4)𝑎𝑛𝑑(6, −7) 6.(0, 12)𝑎𝑛𝑑(−4, 9)7.Usethedistanceformulainordertoclassifythetrianglebelowasequilateral(allsidesthesamelength),isosceles(twosidesthesamelength)orscalene(allsidesdifferentlengths).8.Usethemapbelowtofindtheflyingdistanceinblocksbetweenthetwolandmarks.For#9-12,determineifthequantityinColumnAorColumnBislarger,oriftheyarethesame.9.

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ALGEBRAIANDGEOMETRYSUMMERASSIGNMENT–2020 TOPIC5:SATPREP(ItisnevertooearlytopracticefortheSAT!)SHOWYOURWORKFOREACHPROBLEM1. 2. 3. 4.Questions5and6refertothefollowinginformation

5. 6.