Represented below are the pastures of three neighbors that all surround a triangular watering hole. Each neighbor’s pasture is square as shown in the picture. The side length of the watering hole is given in each picture. The owner of the yellow field thinks the area of his field is the same as the areas of the other two fields combined. For each picture decide whether or not the owner of the yellow field is correct and explain why. 2. 1. 3yd 3yd 5yd 4yd 7yd 5yd 4. 3. 3yd 3yd 3yd 5yd 5yd 3yd 5. 6. 4yd √40 yd 6yd 2yd √32yd 4yd 7. What do you notice about the watering hole when the owner of the yellow field was correct? 9. The area of the yellow pasture is the same size as the area of the other two pastures put together. What is the side length of the watering hole (solve for c)? 3yd c 3yd Name:&________________________________________________&Date:&_____________&Class:&___________& Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6& A.$$Using$the$Pythagorean$Theorem&&& & In¶sailing,&a&towrope&is&used&to&attach&a¶sailer&to&a&boat.& & 1.&What&type&of&triangle&is&formed&by&the&horizontal&distance,&the&vertical&height,&and& the&towrope?&Explain&and&provide&a&picture&to&support&your&claim.& & & & & & 2.&Suppose&the&wind&picks&up&and&the¶sailer&rises&to&50&feet&and&remains&72&feet& behind&the&boat.&Write&an&equation&that&will&help&you&find&how&much&towrope&c&the& parasailer&will&need.&& & & 3.&Solve&the&equation&to&find&the&amount&of&rope&the¶sailer&will&need.&Round&to& the&nearest&foot.&__________& & & & **4.&Suppose&the&towrope&is&300&feet&long&and&the¶sailer&is&200&feet&above&the& water&surface.&Write&an&equation&to&find&the&horizontal&distance&b&behind&the&boat.& & & & B.&Draw&pictures&of&each&of&the&following&relationships&described&to&solve&each& problem.& & 1.&The&janitor&needs&to&fix&a&window&on&the&side&of&the&school&wall.&The&ladder&he&has& is&15&feet&long.&The&window&is&12&feet&high.&How&far&from&the&base&of&the&wall&does& the&ladder&need&to&be&for&him&to&reach&the&window?& & & & & & & 2.&Two&planes&pass&each&other&in&the&air.&They&are&12&miles&apart.&Plane&B&is&10&miles& east&of&Plane&A.&How&many&miles&north&of&Plane&B&is&Plane&A?& & & & & & & 8.G.5%extension% Name:&________________________________________________&Date:&_____________&Class:&___________& Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6& & 3.&A&wire&is&needed&to&stabilize&a&tentYpole.&The&tent&is&12&feet&high&at&the¢er&and& the&bottom&is&a&square&area.&To&stabilize&the&pole,&a&wire&will&stretch&from&the&top&of& the&pole&to&each&corner&of&the&square.&The&flagpole&is&7&feet&from&each&corner&of&the& square.&& & & & & & & & C.&Practice$$ For&each&of&the&following&problems,&draw&or&complete&a&picture&to&help&you&solve.& Then&write&an&equation&that&can&be&used&to&answer&the&question.&Then&solve.& & 1.&Notice&that&the&distance&from&the&building,&the&building&itself,&and&the&ladder,&form& a&right&triangle.&Solve&for&the&length&of&the&ladder,&x.&& & & & & & & & 2.&Mr.&Parsons&wants&to&build&a&new&banister&for&the&staircase&shown&below.&If&the& rise%of&the&stairs&of&a&building&is&5&feet,&and&the&run&of&the&stairs&is&12&feet,&what&will& be&the&length&of&the&new&banister?& & & & & & & % % ! 3.&Taylor&is&waiting&for&her&ride&after&school.&She¬ices&that&her&shadow&is&about&4& feet&long.&With&her&snow&boots&on,&Taylor&is&just&about&6&feet.&What&is&the&distance& from&the&top&of&Taylor’s&head&to&the&tip&of&her&shadow?& & & & & & & 8.G.5%extension% Name:&________________________________________________&Date:&_____________&Class:&___________& Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6& & 4.&McKade&wants&to&go&from&his&house&to&his&grandmother’s&house&(which&is&across&a& field),&but&is&considering&stopping&at&the&gas&station&first.&The&gas&station&is&4&blocks& west&of&McKade’s&house.&Grandmother’s&house&is&3&blocks&north&of&the&gas&station.& & a.&What&is&the&distance&from& McKade’s&house&to&Grandmother’s& if&he&stops&at&the&gas&station?& & & & & b.&What&is&the&distance&McKade& must&travel&if&he&instead&cuts& straight&down&the&field&to&his& Grandmother’s?& & & & & & & 5.&Joylynne&is&walking&home&from&school.&She&walks&two&blocks&north,&and&then&turns& and&walks&3&blocks&east.&How&far&is&Joylynne’s&home&from&school?& & & & & & & & D.&Additional$Practice& If&necessary,&draw&a&picture&to&represent&the&story.&& Write&an&equation&that&can&be&used&to&answer&the&question.&Then&solve.&& 1.&What&is&the&height&of&the&tent?& & & & & & 8.G.5%extension% Name:&________________________________________________&Date:&_____________&Class:&___________& Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6& & 2.&How&high&is&the&wheelchair&ramp?& & & & & & & 3.&Adam&made&a&model&of&a&pyramid&for&a&history&class.&The&diagonal&of&the&square& base&is&18&inches&long.&The&side&of&each&triangular&face&is&15&inches&long.& a.&Draw&the&pyramid&in&the&space&provided:& & & & & & & & b.&What&is&the&height&of&the&model?& & & & & & & & & & & 4.&The&top&part&of&a&circus&tent&is&in&the&shape&of&a&cone.&The&tent&has&a&radius&of&50& feet,&and&the&distance&from&the&top&of&the&tent&to&the&edge&is&61&feet.&How&tall&is&the& top&part&of&the&tent?& & & & & & & & & & & 8.G.5%extension% Name:&________________________________________________&Date:&_____________&Class:&___________& Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6& & Extension$1:$$ Of&the&famous&Seven&Wonders&of&the&Ancient&World&the&Great&Pyramid&of&Khufu& (Cheops)&at&Giza&is&the&only&one&still&standing.&Even&for&modern&men&it&is&amazing& how&this&manYmade&structure&lasted&so&long.& & The&square&base&of&the&pyramid&has&sidelengths&of&about&230&meters.&The&height&of& the&pyramid&when&created&was&roughly&147&meters.&& & The&slant&height&of&a&pyramid&is&the&height&of&each&lateral&face,&or&the&triangles&that& make&up&the&four&sides.&What&is&the&slant&height&of&the&Great&Pyramid?& & & & & & & & & & $ $ $ Extension$2:$ Inigo&Montoya’s&sword&is&very&precious&to&him.&It&is&48&inches&long.&He&is&trying&to& find&a&box&to&place&it&in&to&keep&it&safe&while&he&travels&with&the&Dread&Pirate&Roberts.& The&base&of&the&biggest&box&he&could&find&was&24&inches&by&40&inches.&The&box&itself& is&8&inches&tall.&Can&his&precious&sword&fit?& & & 8.G.5%extension% Name:&___________________________________________________&Date:&_____________&Class:&________& Distance)on)the)Coordinate)Plane)|&Ch&5&Lesson&7& A.))Earlier&this&year,&a&natural&gas&leak&caused&the&“fire&curtain”&to&be&shut&in&the& main&building&at&American&Fork&Junior&High.&&)Mr.&Johnson&was&at&the&old&math& trailer,&investigating&the&leak&when&he&realized&that&he&needed&to&talk&to&Mr.& Schoonover,&who&was&in&the&commons.&&Normally&Mr.&Johnson&would&be&able&to&go& straight&from&the&trailer&to&the&commons.&&However,&because&the&“fire&curtain”&was& shut,&he&had&to&go&outside&the&building&and&in&through&the&front&door&to&get&to&the& commons.&&Would&Mr.&Johnson&walk&farther&by&going&inside&the&building&or&outside& the&building?& ) & & & & & & & & & B& & & & & & & A& & & & & & Above&is&an&aerial&map&of&AFJH,&with&Point&A&being&where&Mr.&Johnson&started,&and& Point&B&representing&the&Commons.& & 1.&Assuming&that&they&walked¶llel&to&the&two&dotted&lines,&draw&the&path&of&the& ARday&students&above.& & 2.&The&distance&from&the&far&side&of&the&road&(horizontal&dotted&line)&to&the&math& building&(Point&A)&is&about&40&meters.&Label%the%scaling%on%each%of%the%dotted%lines.% a.&How&far&away&from&the&far&side&of&the&road&is&the&commons&(Point&B)?& b.&How&far&away&is&the&math&building&from&the&vertical&line?& c.&How&far&away&is&the&commons&from&the&vertical&line?& d.&How&far&did&the&students&have&to&walk&horizontally?&Show%all%work.% e.&How&far&did&the&students&have&to&walk&vertically?&Show%all%work.& & 8.G.8,%8.EE.2% Name:&___________________________________________________&Date:&_____________&Class:&________& Distance)on)the)Coordinate)Plane)|&Ch&5&Lesson&7& 3.&If&the&fire&curtain&was¬&there,&would&they&have&made&it&to&class&sooner&by& walking&straight&from&A&to&B?&Assuming&that&all&walking&rates&are&equal,&explain:% & B.)Evan&was&riding&his&bike&on&a&trail.&A&map&of&the&trail&is&shown.&His&brother&timed& his&ride&from&Point&A&to&Point&B.& & 1.&What&does&the&red&line&represent?& & & 2.&What&do&the&two&blue&lines&represent?& & & 3.&What&type&of&triangle&is&formed&by&the&lines?& & & 4.&How&can&you&find&the&length&of& A C &and& B C without&counting&the&number&of&squares?& & a. What&is&the&length&of& A C ?&Show%all%work.& € € & & b. What&is&the&length&of& B C ?&Show%all%work.& € & & 5.&Write&an&equation&that&you&could&use&to&find&the&length&of& A B ,&and&then&solve.& € & & & € & C.&Think&about&what&you&did&on&parts&A&and&B.&& 1.&How&did&you&find&the&horizontal&differences&in&both&parts&A)and&B?&Was&your& method&the&same?&different?&Describe&below.& & & & & 2.&How&did&you&find&the&vertical&differences&in&both&parts&A)and&B?&Was&your&method& the&same?&different?&Describe&below.& & & & & 3.&What&did&you&use&to&find&the&diagonal&distance&in&both&parts&A&and&B?& & 8.G.8,%8.EE.2% Name:&___________________________________________________&Date:&_____________&Class:&________& Distance)on)the)Coordinate)Plane)|&Ch&5&Lesson&7& D.)Amanda&and&Brooklyn&are&on&a&dance&team.&Their&company&is&doing&a&piece&with& ribbons.&Sadly,&Amanda&and&Brooklyn’s&ribbon&was&torn&when&another&dancer& stepped&on&it.&They&need&to&cut&a&new&one,&but&can’t&remember&the&length!& & When&they&asked&their&instructor,&she&said&that&the&only&information&she&could&give& was&their&locations&during&the&dance.&Amanda&would&be&five&feet&to&the&right&of&the& curtains,&and&two&feet&from&the&front&of&the&stage.&Brooklyn&would&be&14&feet&to&the& right&of&the&curtains,&and&16&feet&from&the&front&of&the&stage.& & 1.&Draw&a&picture&to&represent&Amanda&and&Brooklyn’s&positions&on&the&stage.& & & & & & & & & 2.&Using&your&Pythagorean&Power,&solve&for&the&length&of&the&ribbon!&& & & & & & & & & E.)On&the&grids,&graph&the&ordered&pairs.&Then&find&the&distances&between&the&points.& & 1.&A(3,&0)&and&B(7,&R5)& & & & & & & 2.&C(1,&3)&and&D(R2,&4)& & & & & & & 3.&E(R3,&5)&and&F(2,&7)&& & & 8.G.8,%8.EE.2% Name:&___________________________________________________&Date:&_____________&Class:&________& Distance)on)the)Coordinate)Plane)|&Ch&5&Lesson&7& F.)Formulating)our)Thoughts& & We&found&that&we&can&find&the&distance&between&two&points&by&using&the& Pythagorean&Theorem,& a 2 + b 2 = c 2 .& & However,&instead&of&a,&we&are&squaring&___________________________________________________,& and&instead&of&b,&we&are&squaring&__________________________________________________________.& € & Lastly,&to&get&the&result&of&the&distance&between&the&two&points,&we&are&________________& _________________________________________________________________________________________________ _________________________________________________________________________________________________ ________________________________________________________________________________________________.& & This&process&is&embodied&in&what&is&called&the&Distance)Formula.& & The&Distance)Formula)is&as&follows:& The&distance&d&between&two&points& (x1, y1 ) &and& (x 2 , y 2 ) &is&given&by&the&formula&&&& Let’s&use&it!& & 1.&(3,&0)&and&(7,&R5)& € & & & & 2.&(1,&3)&and&(R2,&4)& & & & & 3.&(R3,&5)&and&(2,&7)& & & & & 4.&(3,&4)&and&(7,&9.5)& & & & & 5.&(2.5,&3.5)&and&(.5,&4)& & & & 6.&(R5,&R3)&and&(R4,&R2)& d = (x 2 − x1 ) 2 + (y 2 − y1 ) 2 .& & € € 8.G.8,%8.EE.2% ! Lines&|!Ch.!5!Lesson!1! ! & !! ! ! ! ! ! ! ! ! ! For&Exercises&1/6,&use&the&figure&at&the&right.&In&the&figure,&& line&m"is¶llel&to&line&n:& List&all&pairs&of&each&type&of&angle.& 1.&!vertical! 2.&!complementary! 3.&!supplementary! 4.&!corresponding! ! 5.&!alternate!interior! ! 6.&!alternate!exterior! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! Use&the&figure&at&the&right&for&Exercises&7/10.& 7.&!Find!the!measure!of!∠2.!Explain!your!reasoning.! ! 8.&!Find!the!measure!of!∠3.!Explain!your!reasoning.! 9.&!Find!the!measure!of!∠4.!Explain!your!reasoning.! 10.!!Find!the!measure!of!∠6.!Explain!your!reasoning.! ! 11.!!ALGEBRA&&Angles!A"and!B"are!corresponding!angles!formed!by!two!parallel! lines!cut!by!a!transversal.!If!m∠A!=!4x"and!m∠B!=!3x"+!7,!find!the!value!of!x.! Explain.! ! 12.!!ALGEBRA&&Angles!G"and!H"are!supplementary!and!congruent.!If!∠G"and!∠H"are! alternate!interior!angles,!what!is!the!measure!of!each!angle?! ! ! ! & ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! For&Exercises&1/12,&use&the&figure&at&the&right.& In&the&figure,&line&m"is¶llel&to&line&n.& Classify&each&pair&of&angles&as&alternate"interior," alternate"exterior,"or"corresponding.& 1.&!∠1!and!∠8! 2.&!∠5!and!∠7! 3.&!∠3!and!∠6! 4.&!∠2!and!∠4! 5.&!∠2!and!∠7! 6.&!∠4!and!∠5! If&m 4&=&122°,&find&each&given&angle&measure.&Justify&your&answer.& 7.&!m∠8! 8.&!m∠5! 9.&!m∠2! 10.!!m∠1! 11.!!m∠6! 12.!!m∠7! For&Exercises&13&and&14,&use&the&figure&at&the&right.& 13.!!List!all!the!angles!congruent!to!the!given!angle.! Explain!your!reasoning.! ! ! ! ! 14.!!List!all!the!angles!congruent!to!∠5.!Explain!your!reasoning.! ! ( Angles'of'Triangles'|'Chapter(5(Lesson(3( ( •((A(triangle'is(formed(by(three(line(segments(that(intersect(only(at(their(endpoints.( ( •((A(point(where(the(segments(intersect(is(a(vertex'of(the(triangle.( ( •((Every(triangle(also(has(three(angles.(The(sum(of(the(measures(of(the(angles(is(180°.( ( Example'1( ( Find(the(value(of(x"in('∆ABC.( " " " " " " ( ( Exercises( Find'the'value'of'x"in'each'triangle.' 1.'( 2.'( 3.'(( 4.'( 5.'( 6.'(( 7.'( 8.'( 9.'(( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Find the value of x in each triangle. 1. 2. 3. 4. 5. 6. Find the missing measure in each triangle with the given angle measures. 7. 45°, 35°, x° 8. 100°, x°, 40° 9. x°, 90°, 16° 10. Find the third angle of a right triangle if one of the angles measures 24°. 11. What is the third angle of a right triangle if one of the angles measures 51°? 12. ALGEBRA Find m∠A in ABC if m∠B = 38° and m∠C = 38°. 13. ALGEBRA In XYZ, m∠Z = 113° and m∠X = 28°, What is m∠Y? Classify the marked triangle in each object by its angles and by its sides. 14. 15. 16. ALGEBRA Find the value of x in each triangle. 17. 18. 19. (
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