geometry unit 2: constructions geometry unit 2: constructions · geometry unit 2: constructions ms....

Geometry Unit2:Constructions

Ms.Talhami 1

GeometryUnit2:Constructions

Name_________________


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HelpfulVocabularyWord Definition/Explanation Examples/HelpfulTips


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HowtoUsetheCompassConstructaCircle


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ConstructaLineSegmentConstructanAngle


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ConstructaTriangleConstructaQuadrilateral


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CopyaSegment(a)Usingyourcompass,placethepointeratPointAandextendituntilreachesPointB.YourcompassnowhasthemeasureofAB.

(b)PlaceyourpointeratA’,andthencreatethearcusingyourcompass.Theintersectionisthesameradii,thusthesamedistanceasAB.YouhavecopiedthelengthAB. Practice1.GivenlinesegmentAB:a)CopyAB

b)ConstructalinesegmentwhosemeasureistwiceAB

2.GivenlinesegmentCD:a)CopyCD

b)ConstructalinesegmentthatistheretimesCD.

c)ConstructalinesegmentthatisequaltoAB+CD

A B

B'A B

A'


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3.Given , ,&AB CD EF .Usethecopysegmentconstructiontocreatethenewlengthslistedbelow.a)3AB

b)CD+EF

c)2CD+AD

d)EF–CD

e)ConstructascalenetriangleusingAB,CD,andEF f)ConstructanisoscelestriangleusingCDasthetwolegsandABasthebase:

A BC DE F


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BisectaSegment(a)Given AB (b)PlaceyourpointeratA,extend

yourcompasssothatthedistanceexceedshalfway.Createanarc.

(c)Withoutchangingyourcompassmeasurement,placeyourpointatBandcreatethesamearc.Thetwoarcswillintersect.LabelthosepointsCandD.

(d)PlaceyourstraightedgeonthepapersothatitformsCD .TheintersectionofCD and AB isthebisectorof AB .

1. Bisect line segment AB and CD a) b)

A

B

A

B

D

C

A

B

M

D

C

A

B


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2. Construct a line segment that is 1 and half times CD:

3. Construct a line segment that is 2 and half times AB:

4. Given AB & CD . Use the midpoint construction to find the midpoint of AB & CD

A B

C

D


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5. Use your midpoint construction to determine the exact length of 14EF

ConstructaMedian

Word Definition/Explanation Examples/HelpfulTips

Median

E F


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Practice1.ConstructamediantoAB

2.ConstructamediantoAB

ExtraPractice13. Given VB -- perform the midpoint construction. This time labeling the two intersection found to be H and K. Draw in , , ,& .VH VK BH BK Also draw HK .

Why is VH = VK? _______________________________________________________________________ Why is BH = BK? _______________________________________________________________________ Why is VH = VK = BH = BK? _______________________________________________________________

V B


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CopyanAngle(a)Givenanangleandaray. (b)Createanarcofanysize,such

thatitintersectsbothraysoftheangle.LabelthosepointsBandC.

(c)CreatethesamearcbyplacingyourpointeratA’.TheintersectionwiththerayisB’.

(d)PlaceyourcompassatpointBandmeasurethedistancefromBtoC.UsethatdistancetomakeanarcfromB’.TheintersectionofthetwoarcsisC’.

(e)Drawtheray ' 'A C (f)Theanglehasbeencopied.

Practice1.Copy∠A

A

A'

C

A

A'

B

B'

C

A

A'

B

C'

B'

C

A

A'

B

C'

B'

C

A

A'

B

o

o

C'

B'

C

A

A'

B


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2.Copy∠B

3. Construct and angle twice ∠B

4. Given ΔABC, construct a copy of it, ΔA’B’C’, by copying angles.

B C

A


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BisectanAngle(a)Givenanangle. (b)Createanarcofanysize,such

thatitintersectsbothraysoftheangle.LabelthosepointsBandC.

(c)Leavingthecompassthesamemeasurement,placeyourpointeronpointBandcreateanarcintheinterioroftheangle.

(d)Dothesameasstep(c)butplacingyourpointeratpointC.LabeltheintersectionD.

(e)Create AD . AD istheanglebisector.

(f) AD istheanglebisector.

Practice1.Bisectthegivenangles:

A

C

A

B

C

A

B

DC

A

B

DC

A

B

oo

DC

A

B


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2.Constructandanglethatis1.5theangle:

ConstructPerpendicularLines


Perpendicular


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ConstructaPerpendicularBisectorofaLineSegment

Practice1.Given AB &CD .ConstructtheperpendicularbisectorsABandCD

A B

C

D

(a)Given AB (b)PlaceyourpointeratA,extendyourcompasssothatthedistanceexceedshalfway.Createanarc.

(c)Withoutchangingyourcompassmeasurement,placeyourpointatBandcreatethesamearc.Thetwoarcswillintersect.LabelthosepointsCandD.

(d)PlaceyourstraightedgeonthepaperandcreateCD .

(e)CD istheperpendicularbisectorof AB .

A

B A

B

D

C

A

B

M

D

C

A

BM

D

C

A

B


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ConstructaPerpendicularLinetoaGivenPointOntheLine(a)Givenapointonaline. (b)PlaceyourpointerapointA.

CreatearcsequaldistantfromAonbothsidesusinganydistance.LabeltheintersectionpointsBandC.

(c)PlaceyourpointeronpointBandextenditpastA.CreateanarcaboveandbelowpointA.

(d)PlaceyourpointeronpointCandusingthesamedistance,createanarcaboveandbelowA.LabeltheintersectionsaspointsDandE.

(e)CreateDE . f)DE isperpendiculartothelinethroughA.

1. Construct a line perpendicular to a given segment through a point on the line.

A C A B C A B

E

D

C A B

E

D

C A B

E

D

C A B


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2. Construct a line perpendicular to a given segment through a point on the line

ConstructaPerpendicularLinetoaGivenPointNotOntheLine(a) Given a point A not on the line.

(b) Place your pointer on point A, and extend It so that it will intersect with the line in two places. Label the intersections points B and C.

(c) Using the same distance, place your pointer on point C and create an arc on the opposite side of point A.

(d) Do the same things as step (c) but placing your pointer on point B. Label the intersection of the two arcs as point D.

(e) Create AD (f) AD is perpendicular to the given line through point A.

A

CB

A

CB

A

D

CB

A

D

CB

A

D

CB

A


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1. Construct a line perpendicular to a given line through a point not on the line.

2. Construct a line perpendicular to a given line through a point not on the line:

ConstructAltitudes


Altitude

B A


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1. Construct an altitude to AB

2. Construct an altitude to AB

ConstructParallelLines(a) Given a point not on the line.

(b) Place your pointer at point B and measure from B to C. Now place your pointer at C and use that distance to create an arc. Label that intersection D.

(c) Using that same distance, place your pointer at point A, and create an arc as shown.

(d) Now place your pointer at C, and measure the distance from C to A. Using that distance, place your pointer at D and create an arc that intersects the one already created. Label that point E.

(e) Create AE . (f) AE is parallel to

B C

A

DB C

A

DB C

A

E

DB C

A E

DB C

A E

DB C

A


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1. Construct a line parallel to a given line through a point not on the line.

2. Construct a line parallel to a given line through a point not on the line.

B A


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ConstructInscribedPolygonsWord Examples Non-Examples

Inscribe

Circumscribe

To inscribe a hexagon in a circle, copy the radius and then copy it around the circumference six times, then connect all points on the circle. To inscribe an equilateral triangle in a circle, copy the radius and then copy it around the circumference six times, then connect every other points on the circle. To inscribe a square in a circle, construct the perpendicular bisector of a diameter then connect the four points on the circle.1. Inscribe a regular hexagon in a circle by construction.


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2. Inscribe an equilateral triangle in a circle by construction.

3. Inscribe a square in a circle by construction.

ConstructPointsofConcurrencyofaTriangle


PointofConcurrency


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PerpendicularBisectorsofaTriangle

AngleBisectorsofaTriangle


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MediansofaTriangle

AltitudesofaTriangle


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AcronymforthePointsofConcurrencyofaTriangleP__________B__________C__________

P__________B__________C__________

A__________B__________I__________

A__________B__________I__________

M__________C__________

M__________C__________

A__________O__________

A__________O__________

PointsofConcurrencyofaTriangleSamplePictures

Circumcenter or Circumscribe perpendicular bisectors

Incenter or Inscribe angle bisectors

Orthocenter

altitudes

Centroid medians


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TakeConstructionsaStepFurther1. Given sides of a rectangle. Construct the rectangle.

2. Given the side of a square. Construct the square.

A B

C

D


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MixedReviewQuestions1. Which illustration shows the correct construction of an angle bisector?

2. Which diagram shows a construction of a 45ο angle?

3. Using a compass and straightedge, construct an equilateral triangle with as a side. Using this

triangle, construct a 30° angle with its vertex at A. [Leave all construction marks.]


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4. The diagram below shows the construction of the bisector of .

Which statement is not true? 1)

m∠EBF) = 1

2m∠ABC

3) m∠EBF) =m∠ABC

2)

m∠DBF) = 1

2m∠ABC

4) m∠DBF) =m∠EBF

5. Based on the construction below, which statement must be true?

1)

m∠ABD = 1

2m∠CBD

3) m∠ABD = m∠ABC

2) m∠ABD =m∠CBD 4)

m∠CBD = 1

2m∠ABD

6. As shown in the diagram below of Δ ABC , a compass is used to find points D and E, equidistant from point A. Next, the compass is used to find point F, equidistant from points D and E. Finally, a straightedge is used to draw AF

u ruu. Then, point G, the intersection of and side of Δ ABC , is

labeled. Which statement must be true?

1) AFu ruu

bisects side BC 3) AFu ruu

⊥BC 2) AF

u ruu bisects ∠BAC

4) Δ ABG Δ ACG


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7. Line segment AB is shown in the diagram below.

Which two sets of construction marks, labeled I, II, III, and IV, are part of the construction of the perpendicular bisector of line segment AB? 1) I and II 2) I and III 3) II and III 4) II and IV

8. One step in a construction uses the endpoints of to create arcs with the same radii. The arcs intersect above and below the segment. What is the relationship of and the line connecting the points of intersection of these arcs? 1) collinear 2) congruent 3) parallel 4) perpendicular

9. The diagram below shows the construction of the perpendicular bisector of

Which statement is not true? 1) AC = CB 2) CB = ½ AB 3) AC = 2AB 4) AC + CB = AB

10. Based on the construction below, which conclusion is not always true?


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11. Use a compass and straightedge to divide line segment AB below into four congruent parts. [Leave all construction marks.]

12. The diagram below illustrates the construction of parallel to through point P. Which statement justifies this construction?

13. Which geometric principle is used to justify the construction below?


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14. The diagram below shows the construction of a line through point P perpendicular to line m.

15. In the accompanying diagram of a construction, what does represent?

16. Which diagram shows the construction of an equilateral triangle? 1) 2) 3) 4)


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17. Which diagram represents a correct construction of equilateral , given side ? 1) 2) 3) 4)

18. On the ray drawn below, using a compass and straightedge, construct an equilateral triangle with a vertex at R. The length of a side of the triangle must be equal to a length of the diagonal of rectangle ABCD.

geometry unit 2: constructions geometry unit 2: constructions · geometry unit 2: constructions ms....

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