Geometric Constructions - Ms. Talhami€¦ · Geometric Constructions (1) Copying a segment (a) Using your compass, place the pointer at Point A and extend it until reaches Point

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Geometric Constructions (1) Copying a segment (a) Using your compass, place the pointer at Point A and extend it until reaches Point B. Your compass now has the measure of AB.

(b) Place your pointer at A’, and then create the arc using your compass. The intersection is the same radii, thus the same distance as AB. You have copied the length AB.

1. Given line segment AB: a) Copy AB

b) Construct a line segment whose measure is twice AB

2. Given line segment CD: a) Copy CD

b) Construct a line segment that is there times CD.

c) Construct a line segment that is equal to AB + CD

A B

B'A B

A'

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3. Given , ,&AB CD EF . Use the copy segment construction to create the new lengths listed below. a) 3AB

b) CD + EF

c) 2CD + AD

d) EF – CD

4. a) Construct an equilateral triangle using AB:

b) Construct an equilateral triangle using CD

A BC DE F

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c) Construct an equilateral triangle using EF d) construct a scalene triangle using AB, CD and EF e) Construct a Isosceles triangle using CD as the two legs and AB as the base:

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1.ConstructanIsoscelestriangleuseABasthelegsandCDasthebase.

2.ConstructatriangleuseAB,CD,andEFasthe3sides.

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(2)Bisectasegment(a)Given AB (b)PlaceyourpointeratA,extend

yourcompasssothatthedistanceexceedshalfway.Createanarc.

(c)Withoutchangingyourcompassmeasurement,placeyourpointatBandcreatethesamearc.Thetwoarcswillintersect.LabelthosepointsCandD.

(d)Placeyourstraightedgeonthe

papersothatitformsCD .The

intersectionofCD and AB isthe

bisectorof 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. Bisect line segment AB and CD a) c)

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

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

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

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

A B

C

D

E F

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Amedianisalinesegmentthatconnectsthemidpointofonesideofatriangleandtheoppositevertex.1.ConstructamediantoAB

2.ConstructamediantoAB

3.ConstructamediantoAC


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5.ConstructamediantoAC


7.ConstructamediantoBC

8.ConstructamediantoBC

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(3) Copy an angle (a) Given an angle and a ray. (b) Create an arc of any size,

such that it intersects both rays of the angle. Label those points B and C.

(c) Create the same arc by placing your pointer at A’. The intersection with the ray is B’.

(d) Place your compass at point B and measure the distance from B to C. Use that distance to make an arc from B’. The intersection of the two arcs is C’.

(e) Draw the ray ' 'A C (f) The angle has been copied.

1. 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. Copy ∠C

4. Construct and angle twice ∠C

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5. Construct and angle twice ∠B

6. Copy ∠C

7. Construct and angle three times ∠D

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8. Given ABC∠ . Make a copy of ABC∠ , ' ' 'A B C∠ .

9. Given DEF∠ . Make a copy of DEF∠ , ' ' 'D E F∠ .

A

B

C

E

B'

D

E

F

E'

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10. Given MN , construct 2.5 MN

11. Given GH , construct 1.75 GH

12. Given ΔABC, construct a copy of it, ΔA’B’C’.

M N

G H

B C

A

A

B'

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13. 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? _______________________________________________________________ What is the most specific name for the quadrilateral VHBK? _______________________________ Will this specific quadrilateral be formed every time using this construction? Yes or No Why or why not… Label the intersection of HK and VB is point M. What is true about VM and BM? ____________________________________________________ What is true about HM and KM? ____________________________________________________ What is the measure of the angle formed at the intersection of HK and VB ? ______________

V B

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1. Construct a Triangle using the following angle and 2 sides

3. Construct a Triangle using the following angles and side.

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3. Construct a Triangle using the following angles and side.

2. Construct a Triangle using the following angle and 2 sides

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(4) Bisect an angle (a) Given an angle. (b) Create an arc of any size,

such that it intersects both rays of the angle. Label those points B and C.

(c) Leaving the compass the same measurement, place your pointer on point B and create an arc in the interior of the angle.

(d) Do the same as step (c) but placing your pointer at point C. Label the intersection D.

(e) Create AD . AD is the angle bisector.

(f) AD is the angle bisector.

1. Bisect the given angles:

A

C

A

B

C

A

B

DC

A

B

DC

A

B

oo

DC

A

B

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2. Construct and angle that is 1.5 the angle:

4. Construct an angle that is 2.5 the angle.

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(5) Construct the perpendicular bisector of a line segment (a) Given AB (b) Place your pointer at A,

extend your compass so that the distance exceeds half way. Create an arc.

(c) Without changing your compass measurement, place your point at B and create the same arc. The two arcs will intersect. Label those points C and D.

(d) Place your straightedge on the paper and createCD .

(e) CD is the perpendicular bisector of AB .

1. Construct the perpendicular bisectors AB and CD

a) b)

A

B A

B

D

C

A

B

M

D

C

A

BM

D

C

A

B

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2. Construct the perpendicular bisectors AB and CD a) c)

3. Given AB & CD . Construct the perpendicular bisectors AB and CD

A B

C

D

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(6) Construct a line perpendicular to a given line through a point not on the line. (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:

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

B A

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AAltitudeisalinewhichpassesthroughavertexofatriangleandmeetstheoppositesideatrightangles.1.ConstructanaltitudetoAB

2.ConstructanaltitudetoAB

3.ConstructanaltitudetoAC


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6.ConstructanaltitudetoBC



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1. Use a compass and a straightedge to construct the following reflections.


Rm

m

A

B

C

Rm

mA

B

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Rm

mA

B

C

Rm

mA

B

C

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6. Determine the Line of Reflection

In trying to find the line of reflection you need to work backwards through the definition of a reflection.

Construct the line of reflection of ΔABC & ΔA’C’B’ How do you know that this is a reflection and not a rotation

Rm

m

A

B

C

A'

B'

C'

C B

A

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7. Determine the Line of Reflection What about this transformation tells you that it must be a reflection and not something else?

Construct the line of reflection of ΔABC & ΔA’C’B’.

8. Determine the Line of Reflection What in this diagram gives us a clue about where the line of reflection is? Construct the line of reflection of ΔABC & ΔA’C’B’.

A'

B'

C'

C B

A

A'

B'

C'

C

B

A

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(7) Construct a line perpendicular to a given segment through a point on the line. (a) Given a point on a line. (b) Place your pointer a point A.

Create arcs equal distant from A on both sides using any distance. Label the intersection points B and C.

(c) Place your pointer on point B and extend it past A. Create an arc above and below point A.

(d) Place your pointer on point C and using the same distance, create an arc above and below A. Label the intersections as points D and E.

(e) CreateDE . f) DE is perpendicular to the line through A.

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

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

AC

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3. Given AB . Construct the perpendicular bisector.

4. Given ∠A, construct the angle bisector, ray AD .

5. Construct a line perpendicular to a given segment through a point:

A B

A

B

A A

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6. Construct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction.

7. Construct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction. a) b)

8. Construct the angle bisector for the below angle but label everything to display where the Rhombus is found in the construction.

G

H

J

A

B

CD

E

F

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1. Given sides of a rectangle. Construct the rectangle. (Hint) We need perpendicular lines through A and through M.

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

A B

C

D

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3. – 6. Use the diagram to complete the relationship.

3.

a)________=________ b)________≅________

4.

a)________=________ b)________≅________

5.

a)________=________ b)________≅________

6.

a)________=________ b)________≅________

7. – 10. Choose which construction matches the diagram. 7.

a)TheMidpointof BC b)⊥linethroughAc)∠bisectord)Copyasegment

8.

a)Copy∠b)⊥bisectorc)∠bisectord)Copyasegment

9.

a)Copy∠b)⊥bisectorc)∠bisectord)Copyasegment

10.

a)TheMidpointof BC b)⊥linethroughAc)∠bisectord)Copyasegment

11. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and then explain why it MUST be a rhombus.

12. If you are told that MN is the perpendicular bisector ofBCwhere point M is on BC . Draw the diagram and completely label it with all known relationships. 5. If you are constructing the perpendicular line through point A (A is on the line), determine the next step. Step #1 – Place compass at point A, and create two intersections B & C on either side of point A. Step #2 – Place compass pointer at point B and extend its measure beyond A and make an arc above and below point A. Step #3 -- __________________________________________________________________________

M

A

Boo

DC

A

B

CB

A

D

CB

A

DC

A

B

C'

B'

C

A

A'

B

C A B

DC

A

B

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(8) Construct a line parallel to a given line through a point not on the line. (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

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

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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B A

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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.

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

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3. Inscribe a square in a circle by construction.

4. Inscribe a regular hexagon in a circle by construction.

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

6. Inscribe a square in a circle by construction.

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Circumcenter or Circumscribe perpendicular bisectors

Incenter or Inscribed angle bisectors

Orthocenter altitudes

Centroid medians

1. Circumscribe a circle about each triangle

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2. Inscribe a circle in each triangle.

3. Locate the orthocenter of each triangle.

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4. Locate the centroid of each triangle.


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46



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48



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Regents Questions 1. Which illustration shows the correct construction of an angle bisector?

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

3. Construct the angle bisector of the given angle.

4. On the diagram below, use a compass and straightedge to construct the bisector of . [Leave all construction marks.]

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5. Using a compass and straightedge, construct the angle bisector of shown below. [Leave all construction marks.]

6. On the diagram below, use a compass and straightedge to construct the bisector of . [Leave all construction marks.]

7. Using only a ruler and compass, construct the bisector of angle BAC in the accompanying diagram.

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8. Using a compass and straightedge, construct the bisector of . [Leave all construction marks.]

9. 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.] 10. 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

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2)

m∠DBF) = 1

2m∠ABC

4) m∠DBF) =m∠EBF

11. A straightedge and compass were used to create the construction below. Arc EF was drawn from point B, and arcs with equal radii were drawn from E and F.

Which statement is false? 1) m∠ABD =m∠DBC 3) 2(m∠DBC) =m∠ABC 2)

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(m∠ABC) = m∠ABD 4) 2(m∠ABC) =m∠CBD

12. 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

12. A student used a compass and a straightedge to construct CE in Δ ABC as shown below.

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Which statement must always be true for this construction? 1) ∠CEA ≅∠CEB 3) AE ≅BE 2) ∠ACE ≅ ∠BCE 4)

EC ≅ AC

14. 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

1. 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

2. 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

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3) parallel 4) perpendicular

4. 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

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

6. Using a compass and straightedge, construct the perpendicular bisector of . [Leave all construction marks.]

7. Using only a compass and a straightedge, construct the perpendicular bisector of and label it c. [Leave all construction marks.]

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8. Using a compass and straightedge, construct the perpendicular bisector of shown below. Show all construction marks.

9. On the diagram of shown below, use a compass and straightedge to construct the perpendicular bisector of . [Leave all construction marks.]

10. 10. Using a compass and straightedge, construct the perpendicular bisector of side in shown below. [Leave all construction marks.]

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

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

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

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3. The diagram below shows the construction of through point P parallel to .

4. The diagram below shows the construction of line m, parallel to line , through point P.

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

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

7. Using a compass and straightedge, construct a line that passes through point P and is perpendicular to line m. [Leave all construction marks.]

8. 8. Using a compass and straightedge, construct the line that is perpendicular to and that passes through point P. Show all construction marks.

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9. 9. Using a compass and straightedge, construct a line perpendicular to through point P.

[Leave all construction marks.]

10. 10. Using a compass and straightedge, construct a line perpendicular to line through point P. [Leave all construction marks.]

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

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

3. On the line segment below, use a compass and straightedge to construct equilateral triangle ABC. [Leave all construction marks.]

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4. Using a compass and straightedge, and below, construct an equilateral triangle with all sides congruent to . [Leave all construction marks.]

5. Using a compass and straightedge, on the diagram below of , construct an equilateral triangle with as one side. [Leave all construction marks.]

6. 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.

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Documents

Geometric Constructions - Ms. Talhami€¦ · Geometric Constructions (1) Copying a segment (a) Using your compass, place the pointer at Point A and extend it until reaches Point