Chapter 5 Review Relaonships within Triangleschristensen60.weebly.com/uploads/5/4/3/4/54343837/... · 2019-12-06 · 5.1 Midsegment Theorem and Coordinate Proof 5.2 Use Perpendicular

Chapter5ReviewRela/onshipswithinTriangles

5.1MidsegmentTheoremandCoordinateProof5.2UsePerpendicularBisectors5.3UseAngleBisectorsofTriangles5.4UseMediansandAl/tudes5.5UseInequali/esinaTriangle5.6Inequali/esinTwoTrianglesandIndirectProof

CHAPTER5:Rela/onshipswithinTriangles

UsingProper,esofSpecialSegmentsinTriangles

Specialsegment Proper/estoremember

Midsegment Paralleltosideoppositeitandhalfthelengthofsideoppositeit

PerpendicularBisector

Concurrentatthecircumcenter,whichis:•equidistantfrom3ver/cesofΔ•centerofcircumscribedcirclethatpassesthrough3ver/cesofΔ



Anglebisector Concurrentattheincenter,whichis:•equidistantfrom3sidesofΔ•centerofinscribedcirclethatjusttoucheseachsideofΔ

Median(connectsvertextomidpointofoppositeside)

Concurrentatthecentroid,whichis:•locatedtwothirdsofthewayfromvertextomidpointofoppositeside•balancingpointofΔ



Al/tude(perpendiculartosideofΔthroughoppositevertex)

ConcurrentattheorthocenterUsedinfindingarea:Ifbislengthofanysideandhislengthofal/tudetothatside,thenA=½bh.

UsingTriangleInequali,estoDetermineWhatTrianglesarePossible

SumoflengthsofanytwosidesofaΔisgreaterthanlengthofthirdside.

AB + BC > AC AC + BC > AB AB + AC > BC


InaΔ,longestsideisoppositelargestangleandshortestsideisoppositesmallestangle. If AC > AB > BC, then

m∠B > m∠ C > m∠ A.If m∠B > m∠ C > m∠ A,then AC > AB > BC.


IftwosidesofaΔare≅totwosidesofanotherΔ,thentheΔwithlongerthirdsidealsohaslargerincludedangle.

If BC > EF,then m∠A > m∠ D.If m∠A > m∠D,then BC > EF.

ExtendingMethodsforJus,fyingandProvingRela,onships

Coordinateproofusesthecoordinateplaneandvariablecoordinates.Indirectproofinvolvesassumingtheconclusionisfalseandthenshowingthattheassump/onleadstoacontradic/on.

REVIEWKEYVOCABULARY:

•midsegmentofatriangle•coordinateproof•perpendicularbisector•equidistant•pointofconcurrency•circumcenter•incenter•medianofatriangle

REVIEWKEYVOCABULARY:

•centroid•al/tudeofatriangle•orthocenter•indirectproof

1.Copyandcomplete:A______________________isasegment,ray,line,orplanethatisperpendiculartoasegmentatitsmidpoint.

VOCABULARYEXERCISESperpendicularbisector

VOCABULARYEXERCISES2.Explainhowtodrawacirclethatiscircumscribedaboutatriangle.Whatisthecenterofthecirclecalled?Describeitsradius.

Findtheintersec/onofthreeperpendicularbisectorsofthetriangle.Usingthispointasthecenterofthecircle,drawacirclewhoseradiusisthedistancefromthepointtoanyofthever/ces;circumcenter;thedistancefromthecircumcentertoanyofthever/cesofthetriangle.

3.Incenter A.Thepointofconcurrencyofthemediansofatriangle

4.Centroid B.Thepointofconcurrencyoftheanglebisectorsofatriangle

5.Orthocenter C.Thepointofconcurrencyoftheal/tudesofatriangle

VOCABULARYEXERCISESInExercises3–5,matchthetermwiththecorrectdefini/on.

A

B

C

5.1MIDSEGMENTTHEOREMANDCOORDINATEPROOF

DE = 12AC

AC = 2DE = 2(51) = 102

BytheMidsegmentThm:

In the diagram, DE is a midsegment ofΔABC. Find AC.


EF = 12AB

EF = 12(72) = 36

6.IfAB=72,findEF.

Use the diagram where DF and FE are a midsegments of ΔABC.


DF = 12BC = EC

DF = 45 = EC

7.IfDF=45,findEC.

Use the diagram where DF and FE are a midsegments of ΔABC.


8. Graph ΔPQR, with vertices P(2a,2b), Q(2a,0), and O(0,0). Find thecoordinates of midpoint S of PQ and midpoint T of QO. Show ST || PO.

M = x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟

Slope = y2 − y1x2 − x1

SPQ = 2a + 2a2

, 2b + 02

⎛⎝⎜

⎞⎠⎟ =

4a2, 2b2

⎛⎝⎜

⎞⎠⎟ = 2a,b( )

TQO = 2a + 02

, 0 + 02

⎛⎝⎜

⎞⎠⎟ =

2a2, 02

⎛⎝⎜

⎞⎠⎟ = a,0( )

SlopeST =y2 − y1x2 − x1

= b − 02a − a

= ba

SlopeOP =y2 − y1x2 − x1

= 2b − 02a − 0

= baSlopeST = SlopeOP ⇒||

5.2USEPERPENDICULARBISECTORSUsethediagramattherighttofindXZ

5x − 5 = 3x + 32x = 8x = 4XZ = 5x − 5 = 5(4)− 5 = 15

WZ! "##

is the perpendicular bisectorof XY .

5.2USEPERPENDICULARBISECTORS

BA and BC, AD and DC

In the diagram, BD! "##

is the perpendicular bisector of AC.

9.Whatsegmentlengthsareequal?


20 = 7x −1535 = 7xx = 5



10.Whatisthevalueofx?


AB = 6x − 5AB = 6(5)− 5AB = 30 − 5 = 25



11.FindAB.

5.3USEANGLEBISECTORSOFTRIANGLES

a2 + b2 = c2

242 + NM 2 = 302

576 + NM 2 = 900NM 2 = 324

NM = 324 = 18 = NL

In the diagram, N is the incenter of ΔXYZ. Find NL.

Use the Pythagorean Theorem to find NM in ΔNMY .

5.3USEANGLEBISECTORSOFTRIANGLES

RD = x = SD = TD = 5x = 5

Point D is the incenter of the triangle. Find the value of x.

5.3USEANGLEBISECTORSOFTRIANGLESPoint D is the incenter of the triangle. Find the value of x.

CE2 + ED2 = CD2

202 + ED2 = 252

400 + ED2 = 625ED2 = 225

ED = 225 = 15 = GD = xx = 15

5.4USEMEDIANSANDALTITUDESThe vertices of ΔABC are A(−6,8), B(0,−4), andC(−12, 2). Find the coordinates of its centroid P.

MCB =x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟ =

−12 + 02

, 2 + −42

⎛⎝⎜

⎞⎠⎟ = −6,−1( )

AM = 8 − (−1) = 9

AP = 23AM = 2

3⋅9 = 6

P = (−6,8 − 6) = (−6,2)

5.4USEMEDIANSANDALTITUDESFind the coordinates of the centroid D of ΔRST .

MST =x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟ =

2 + 22, 2 + −22

⎛⎝⎜

⎞⎠⎟ = 2,0( )

RM = 2 − (−4) = 6

RD = 23RM = 2

3⋅6 = 4

D = (−4 + 4,0) = (0,0)

14.R(-4,0),S(2,2),T(2,-2)

5.4USEMEDIANSANDALTITUDESFind the coordinates of the centroid D of ΔRST .

MRT =x1 + x22

, y1 + y22

⎛⎝⎜

⎞⎠⎟ =

−6 + 22

, 2 + 42

⎛⎝⎜

⎞⎠⎟ = −2,3( )

SM = 6 − (3) = 3

SD = 23SM = 2

3⋅3= 2

D = (−2,6 − 2) = (−2,4)

15.R(-6,2),S(-2,6),T(2,4)

5.4USEMEDIANSANDALTITUDES

Point Q is the centroid of ΔXYZ.

QN = 13XN

3QN = XN3(3) = 9 = XN

XQ = 23XN

XQ = 23(9) = 6

16.FindXQ.


Point Q is the centroid of ΔXYZ.

XM = 12XY

XM = 12(7)

XM = 72= 3.5

17.FindXM.


Draw an obtuse ΔABC. Draw its threealtitudes. Then label its orthocenter D.

5.5USEINEQUALITIESINATRIANGLEAtrianglehasonesideoflength9andanotheroflength14.Describethepossiblelengthsofthethirdside.

Letxrepresentthelengthofthethirdside.DrawdiagramsandusetheTriangleInequalityTheoremtowriteinequali/esinvolvingx.

9 + x >14x > 5

9 +14 > x23> x

Thelengthofthethirdsidemustbegreaterthan5andlessthan23.

5.5USEINEQUALITIESINATRIANGLEDescribethepossiblelengthsofthethirdsideofthetrianglegiventhelengthsoftheothertwosides.

19)4inches,8inches

4 + x > 8x > 4

8 + 4 > x12 > x

4 in < l <12 in


20).6meters,9meters

6 + x > 9x > 3

9 + 6 > x15 > x

3 m < l <15 m


21)12feet,20feet

12 + x > 20x > 8

20 +12 > x32 > x

8 ft < l < 32 ft

5.5USEINEQUALITIESINATRIANGLEListthesidesandtheanglesinorderfromsmallesttolargest.

RQ,PR,PQ∠P,∠Q,∠R


LM ,MN ,LN∠N ,∠L,∠M


AB,AC,BC∠C,∠B,∠A

5.6INEQUALITIESINTWOTRIANGLESANDINDIRECTPROOF

Because 27! > 23!, m∠GEF > m∠GED. You are given thatDE ≅ FE and you know that EG ≅ EG. Two sides of ΔGEFare congruent to two sides of ΔGED and the includedangle is larger so, by the Hinge Theorem, FG > DG.

How does the length of DG compare to the length of FG?


m∠BAC > m∠DAC.

Copyandcompletewith<,>,or=.


LM = KN .

Copyandcompletewith<,>,or=.


27.ArrangestatementsA–Dincorrectordertowriteanindirectproofofthestatement:Iftwolinesintersect,thentheirintersec=onisexactlyonepoint.GIVEN:Intersec/nglinesmandnPROVE:Theintersec/onoflinesmandnisexactlyonepoint.

A.ButthiscontradictsPostulate5,whichstatesthatthroughanytwopointsthereisexactlyoneline.B.Thentherearetwolines(mandn)throughpointsPandQ.C.Assumethattherearetwopoints,PandQ,wheremandnintersect.D.Itisfalsethatmandncanintersectintwopoints,sotheymustintersectinexactlyonepoint.


27.ArrangestatementsA–Dincorrectordertowriteanindirectproofofthestatement:Iftwolinesintersect,thentheirintersec=onisexactlyonepoint.GIVEN:Intersec/nglinesmandnPROVE:Theintersec/onoflinesmandnisexactlyonepoint.

C.Assumethattherearetwopoints,PandQ,wheremandnintersect.B.Thentherearetwolines(mandn)throughpointsPandQ.A.ButthiscontradictsPostulate5,whichstatesthatthroughanytwopointsthereisexactlyoneline.D.Itisfalsethatmandncanintersectintwopoints,sotheymustintersectinexactlyonepoint.

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Chapter 5 Review Relaonships within Triangleschristensen60.weebly.com/uploads/5/4/3/4/54343837/... · 2019-12-06 · 5.1 Midsegment Theorem and Coordinate Proof 5.2 Use Perpendicular