Geometry Workbook 1

Geometry Workbook 1:

Vocabulary, Geometric Objects and Their Symbols, and

Geometric Constructions

Student Name __________________________________________

STANDARDS:

G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

SKILLS:

I will be able to define any geometric term using words, diagrams, and notation.

I will be able to explain what the undefined terms are and why they are undefined.

I will be able to use geometric concepts to establish a rationale for the steps/procedures used in

performing a construction.

I will be able to use a compass and straightedge to create the following constructions. o Copy a given line segment. o Copy a given angle. o Bisect a line segment. o Bisect an angle. o Construct a line perpendicular to a given line through a point on that line. o Construct a line perpendicular to a given line through a point not on that line. o Construct the perpendicular bisector of a line segment. o Construct a line parallel to a given line through a point not on the line. o Construct geometric figures using a variety of tools and methods.

I will be able to construct the following: o An equilateral triangle inscribed in a circle. o A square inscribed in a circle. o A regular hexagon inscribed in a circle

I will be able to construct the following: o An equilateral triangle o A square o A regular hexagon

Notes:

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G.CO.D.12 STUDENT NOTES & PRACTICE WS #1– geometrycommoncore.com 1

1. Copying a segment

(a) Using your compass, place the pointer at Point A and extend it until it 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 radius, thus the same distance as AB. You have copied the length AB.

NYTS (Now You Try Some)

Copy the given segment.

Create the length 3AB

2. Bisect a 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 so that it forms𝐶𝐷 ⃡ . The

intersection of 𝐶𝐷 ⃡ and 𝐴𝐵̅̅ ̅̅ is the

bisector of 𝐴𝐵̅̅ ̅̅ .

(e) I labeled it M, because it is the midpoint of 𝐴𝐵̅̅ ̅̅ .

A B

B'A B

A'

A B A'

A B A'

A

B

A

B

D

C

A

B

M

D

C

A

B M

A

B



Bisect the segment (find the midpoint).

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.


Copy the given angle.

A

B C

D

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

A A'


4. 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


Find the parallel line though the 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

G.CO.D.12 GUIDED PRACTICE WS #1– geometrycommoncore.com 1

1. Copying a segment

Copy the given segment.

Create the length 3AB

Create the length AB – CD

2. Bisect a segment (find the midpoint)

3. Copy an angle

A B A'

A B A'

A B

C D

A

B C

D

A A'


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

G.CO.D.12 WORKSHEET #1 – geometrycommoncore.com NAME: ______________________ 1

1. Convert the mathematical symbols to words.

a) AB _______________________ b) AB _______________________

c) AB _______________________ d) AB _______________________

e) ABC _______________________ f) m ABC _______________________

2. Which geometric instrument would I use to measure the length of a segment, the compass or the

straightedge? Explain your answer.

3. What is the difference between drawing and constructing something? So for example, what is the

difference between drawing a perpendicular line and constructing a perpendicular line?

4. A student has done the following construction. What was this student attempting to construct? Is there

more than one thing that the student could be constructing? Explain.

5. After learning the midpoint construction, Sally realizes that she could determine one-fourth the length of

a segment. How could she do this? Explain & Diagram.

6. When given AB & CD , a student uses her compass to measure them and then construct a new length

EF. What is the exact length of EF ?

F

C D

A B

E

D

C

A

B

G.CO.D.12 WORKSHEET #1 – geometrycommoncore.com 2

7. A teacher instructs the class to construct the midpoint of a segment. Jeff pulls out his ruler and measures

the segment to the nearest millimeter and then divides the length by two to find the exact middle of the

segment. Has he done this correctly?

8. What is the difference between CD and CD?

9. Give three possible correct names for the given angle.

10. When do we use = and when do we use ?

11. What does it mean to bisect something?

12. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus. Can you

find it? Explain why it MUST be a rhombus.

1C

A

B

D

C

A

B

G.CO.D.12 ACTIVITY #1 – geometrycommoncore.com NAME: _________________________ 1

COPYING A SEGMENT

1. Given , ,&AB CD EF . Use the copy segment

construction to create the new lengths listed below.

3AB

CD + EF

2CD + 1AB

EF - CD

CONSTRUCTING A MIDPOINT

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

3. Use your midpoint construction to determine the exact length of 1

4EF

A BC DE F

A B

C

D

E F

G.CO.D.12 ACTIVITY #1 – geometrycommoncore.com 2

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

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

A

B

C

E

B'

D

E

F

E'


PRACTICE - CONSTRUCTION BASICS #1

1. Given MN , construct 2.5 MN

2. Given GH , construct 1.75 GH

3. Given ABC, construct a copy of it, A’B’C’.

M N

G H

B C

A

A

B'


4. Given ABC, can you think of a way to create a line parallel to AB through point C?

(Hint: How could copying an angle help you?)

5. Create a parallel line to 𝐷𝐸 ⃡ through point F.

BC

A

D F

E


1. 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 create 𝐶𝐷 ⃡ . (e) 𝐶𝐷 ⃡ is the perpendicular bisector of 𝐴𝐵̅̅ ̅̅ .


Construct the perpendicular bisector.

A

BA

B

D

C

A

B

M

D

C

A

B

M

D

C

A

B

A

B CD


2. 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 at point A. Create arcs equidistant 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) Create DE . f) DE is perpendicular to the line through A.


Construct the perpendicular line through a point on the line.

AC A B C A B

E

D

C A B

E

D

C A B

E

D

C A B


3. 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 place 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.


Construct the perpendicular line through a point not on the line.

A

CB

A

CB

A

D

CB

A

D

CB

A

D

CB

A


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 place your pointer at point C. Label the intersection D.

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

(f) AD is the angle bisector.


Bisect the angle.

A

C

A

B

C

A

B

D

C

A

B

D

C

A

B

o

oD

C

A

B


1. Construct the perpendicular bisector of a line segment

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

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

A

B CD


4. Bisect an angle

G.CO.D.12 WORKSHEET #2 – geometrycommoncore.com NAME: ______________________ 1

1. Use the diagram to complete the relationship.

a) AM = ________

b) AM ________

c) m CAD = ________

d) CAD ________

e) AB = ________

f) AB ________

g) AC = ________

h) AC ________

2. Choose which construction matches the diagram.

a) The Midpoint of BC

b) line through A

c) bisector d) Copy a segment

a) Copy

b) bisector


a) Copy

b) bisector


a) The Midpoint of BC

b) line through A


3. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus. Can you

find it? Explain why it MUST be a rhombus.

4. You are told that MN is the perpendicular bisector of BC where 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

Bo

oD

C

A

B

CB

A

D

CB

A

D

C

A

BC'

B'

C

A

A'

B

C A B

D

C

A

B


REVIEW OF ALGEBRA…… APPLIED TO GEOMETRY

Using a straight-edge and compass we have copied segments and constructed midpoints in the general

plane. If we move to a coordinate plane we are able to measure segments and determine midpoint

locations using the distance formula and the midpoint formula. Let us use these tools to solve a few

problems.

DISTANCE FORMULA MIDPOINT FORMULA

2 2

2 1 2 1x x y y 1 2 1 2, ,2 2

m m

x x y yx y

6. Determine the distance between the two points.

a) A(3,5) B(7,8) b) D(-3,4) E(2,-8)

c) R(0,6) S(-4,-6) d) T(-2,-4) S(-10,-6)

7. Determine the midpoint of the following segments.

a) A (6,-1) B (2,-5) b) A (7,-2) B (-12,4) c) A 2 3

,3 2

B 1 1

,3 2

8. Circle G has a diameter AB where A (2,3) and B(-4,-5). Determine the center of the circle and its radius.

G.CO.D.12 ACTIVITY #2 – geometrycommoncore.com NAME: ________________________ 1

Constructing the Perpendicular Bisector (a line through the midpoint of a segment).

1. Given AB . Use the midpoint construction to construct the perpendicular bisector.

Construct the perpendicular line THROUGH A POINT ON THE LINE. 2. Work backwards from the midpoint construction.

Construct the perpendicular line THROUGH A POINT not on THE LINE. 3. Work backwards through the midpoint construction.

A B

A

B

A

C

B A


Construct the angle bisector.

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

5. Given sides of a rectangle. Construct the rectangle. Hint - We need perpendicular lines through A and through B.

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

A

A

A B

C

D

G.CO.D.12 ACTIVITY #3 – geometrycommoncore.com NAME: ________________________ 1

THE HIDDEN RHOMBUS

Why do these constructions work?

Construct a Midpoint Construct a Perpendicular Line

Through a Point Not on the Line Construct a Perpendicular Line

Through a Point On the Line

Construct an Angle Bisector

1. When performing these constructions you kept your compass at a fixed length. This creates distances that are equal - thus the 4 equal sides of a rhombus. a) Use a ruler and draw in the missing sides of the rhombus in each of these constructions. b) Shade the rhombus with your pencil.

2. In the first three constructions we needed to create lines that were perpendicular.

Which property of a rhombus do you think we are using to accomplish this goal?

Opposite sides are || Diagonals bisect each other

Opposite sides are Diagonals are

Opposite angles are Diagonals are Bisectors

Consecutive ’s = 180

3. In the angle bisector construct we need to divide an angle into two equal parts.

Which property of a rhombus do you think we are using to accomplish this goal?

Opposite sides are || Diagonals bisect each other

Opposite sides are Diagonals are

Opposite angles are Diagonals are Bisectors

Consecutive ’s = 180


4. 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 as 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

G.CO.D.13 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com 1

1. The construction of an inscribed equilateral triangle.

(A) Given Circle A (B) Create a diameter BC (C) Create a circle at C with radius

AC . Label the two intersections D and E.

(D) Create BD , BE & ED (E) The inscribed Equilateral Triangle

NYTS (Now You Try Some) Construct an inscribed Equilateral Triangle.

A

B

A

C

B

E

D

A

C

B

E

D

A

C

B

E

D

A


2. The construction of an inscribed square.

(A) Given Circle A (B) Create a diameter BC (C) Construct a perpendicular line

to BC through A.

(D) Create BD , DC , CE & EB (E) The inscribed Square

NYTS (Now You Try Some) Construct an inscribed Square.

A

B

A

C

E

D

B

A

C

E

D

B

A

C

E

D

B

A

C


3. The construction of an inscribed hexagon.

(A) Given Circle A (B) Create a diameter BC (C) Create a circle at C with radius

AC . Label the two intersections D and E.

(D) Create a circle at B with radius

BA . Label the two intersections F and G.

(E) Create CD , DF , FB , BG ,

GE & EC

(F) The inscribed Hexagon

NYTS (Now You Try Some) Construct an inscribed Hexagon.

A

B

A

C

B

E

D

A

C

G

F

D

E

B

A

C

G

F

D

E

B

A

C

G

F

D

E

B

C

G.CO.D.13 GUIDED PRACTICE WS #1 – geometrycommoncore.com 1

1. The construction of an inscribed equilateral triangle.

2. The construction of an inscribed square.

3. The construction of an inscribed hexagon.

G.CO.D.13 WORKSHEET #1 – geometrycommoncore.com NAME: _____________________ 1

1. Determine whether the relationships is INSCRIBED or CIRCUMSCRIBED.

a) The triangle is _____________. b) The hexagon is _____________ c) The circle is _______________

d) The hexagon is _____________ e) The circle is _______________ f) The triangle is ______________

2. Jeff uses his compass to make a cool design. He just keeps creating congruent circles… over and over…

a) Find a regular hexagon (shade it in) b) Find a different regular hexagon (shade it in)

c) Find an equilateral triangle (shade it in) d) Find a different equilateral triangle (shade it in)


3. The inscribed equilateral triangle has a central angle of 120 because 360 / 3 = 120, an inscribed square

has a central angle of 90 because 360 / 4 = 90. The central angle of a decagon is 36 because 360 / 10 =

36. Use this information and a compass to create an inscribed decagon.

(A decagon has 10 sides.)

4. Construct the requested inscribed polygons.

a) Construct an equilateral triangle inscribed in the provided circle using your compass and straightedge.

b) Construct a square inscribed in the provided circle using your compass and straightedge.

36°


5. Construct the requested inscribed polygons.

a) Construct a regular hexagon inscribed in the provided circle using your compass and straightedge.

b) Construct a regular octagon inscribed in the provided circle using your compass and straightedge.

Hint: The central angle is 45, half of

the square’s central angle of 90.

G.CO.D.13 WORKSHEET #2 – geometrycommoncore.com Name: _________________________ 1

Review the basic constructions.

1. Copying Segments a) 4AB – CD

b) 2.75CD + AB

2. Midpoint Construction

a) Given diameter AB, find the center

b) Which is larger AB or ½ CD? ____________

c) Construct the midpoint of AB

d) 1

3CD GH . Construct GH .

A B C D

B

A

Copy them here to compare them

A

B

C

D

A

B

C

D

G


3. Copying an Angle & Parallel Line Construction

a) Create A’B’C’ by copying ABC. b) Create D’E’F’ by copying DEF.

c) Create A’ by copying Copy A d) Create E’ such that mE’ = 2(mE)

4. Angle Bisector Construction

a) Construct the angle bisector of A. b) Create DBC such that mDBC = ½ (mABC)

B

A

C

B'

E

D

F

E'

A

A'

E

E'

A

B

A

C


5. Perpendicular Constructions

Construct the following.

a) the perpendicular bisector of AD b) a perpendicular line to 𝐴𝐸 ⃡ through E

c) the perpendicular line to 𝐷𝐴 ⃡ through C

6. Inscribed Polygons

a) Inscribed Square b) Inscribed Equilateral Triangle c) Inscribed Hexagon

A

C

E

D

Geometry Workbook 1

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