Use$the$Pythagorean$Theorem

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&parasailing,&a&towrope&is&used&to&attach&a&parasailer&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.&
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2.&Suppose&the&wind&picks&up&and&the&parasailer&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.&&
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3.&Solve&the&equation&to&find&the&amount&of&rope&the&parasailer&will&need.&Round&to&
the&nearest&foot.&__________&
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**4.&Suppose&the&towrope&is&300&feet&long&and&the&parasailer&is&200&feet&above&the&
water&surface.&Write&an&equation&to&find&the&horizontal&distance&b&behind&the&boat.&
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B.&Draw&pictures&of&each&of&the&following&relationships&described&to&solve&each&
problem.&
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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?&
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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?&
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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&center&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.&&
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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.&&
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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?&
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!
3.&Taylor&is&waiting&for&her&ride&after&school.&She&notices&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?&
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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?&
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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?&
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D.&Additional$Practice&
If&necessary,&draw&a&picture&to&represent&the&story.&&
Write&an&equation&that&can&be&used&to&answer&the&question.&Then&solve.&&
1.&What&is&the&height&of&the&tent?&
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8.G.5%extension%
Name:&________________________________________________&Date:&_____________&Class:&___________&
Use$the$Pythagorean$Theorem$|&Chapter&5&Lesson&6&
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2.&How&high&is&the&wheelchair&ramp?&
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3.&Adam&made&a&model&of&a&pyramid&for&a&history&class.&The&diagonal&of&the&square&
base&is&18&inches&long.&The&side&of&each&triangular&face&is&15&inches&long.&
a.&Draw&the&pyramid&in&the&space&provided:&
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b.&What&is&the&height&of&the&model?&
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4.&The&top&part&of&a&circus&tent&is&in&the&shape&of&a&cone.&The&tent&has&a&radius&of&50&
feet,&and&the&distance&from&the&top&of&the&tent&to&the&edge&is&61&feet.&How&tall&is&the&
top&part&of&the&tent?&
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8.G.5%extension%
Name:&________________________________________________&Date:&_____________&Class:&___________&
Use$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.&&
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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?&
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$
$
$
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?&
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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?&
)
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B&
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A&
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Above&is&an&aerial&map&of&AFJH,&with&Point&A&being&where&Mr.&Johnson&started,&and&
Point&B&representing&the&Commons.&
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1.&Assuming&that&they&walked&parallel&to&the&two&dotted&lines,&draw&the&path&of&the&
ARday&students&above.&
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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.&
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8.G.8,%8.EE.2%
Name:&___________________________________________________&Date:&_____________&Class:&________&
Distance)on)the)Coordinate)Plane)|&Ch&5&Lesson&7&
3.&If&the&fire&curtain&was&not&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.&
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1.&What&does&the&red&line&represent?&
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2.&What&do&the&two&blue&lines&represent?&
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3.&What&type&of&triangle&is&formed&by&the&lines?&
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4.&How&can&you&find&the&length&of& A C &and& B C
without&counting&the&number&of&squares?&
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a. What&is&the&length&of& A C ?&Show%all%work.&
€
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b. What&is&the&length&of& B C ?&Show%all%work.&
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5.&Write&an&equation&that&you&could&use&to&find&the&length&of& A B ,&and&then&solve.&
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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.&
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2.&How&did&you&find&the&vertical&differences&in&both&parts&A)and&B?&Was&your&method&
the&same?&different?&Describe&below.&
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3.&What&did&you&use&to&find&the&diagonal&distance&in&both&parts&A&and&B?&
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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.&
&
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2.&Using&your&Pythagorean&Power,&solve&for&the&length&of&the&ribbon!&&
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E.)On&the&grids,&graph&the&ordered&pairs.&Then&find&the&distances&between&the&points.&
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1.&A(3,&0)&and&B(7,&R5)&
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2.&C(1,&3)&and&D(R2,&4)&
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3.&E(R3,&5)&and&F(2,&7)&&
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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!&
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1.&(3,&0)&and&(7,&R5)& €
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2.&(1,&3)&and&(R2,&4)&
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3.&(R3,&5)&and&(2,&7)&
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4.&(3,&4)&and&(7,&9.5)&
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5.&(2.5,&3.5)&and&(.5,&4)&
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6.&(R5,&R3)&and&(R4,&R2)&
d = (x 2 − x1 ) 2 + (y 2 − y1 ) 2 .&
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8.G.8,%8.EE.2%
!
Lines&|!Ch.!5!Lesson!1!
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For&Exercises&1/6,&use&the&figure&at&the&right.&In&the&figure,&&
line&m"is&parallel&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!
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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?!
!
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For&Exercises&1/12,&use&the&figure&at&the&right.&
In&the&figure,&line&m"is&parallel&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.!
!
!
!
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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.
(