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GEOMETRY
2013-2014
Geometry Tools
August 27, 2013
• Toolbox
• Automaticity
STANDARDS
Trimester 1
Day 1-----G.CO.13
Equilateral Triangle
& Regular Hexagon
Day 2 & 3 -----FAL:
Having Kittens
Day 8 -----G.CO.12
Perpendiculars and
G.CO.13 Inscribed
Square
Day 9-----G.CO.12 Parallel
Lines
Day 3 & 4---G.CO.12
Copy Segments &
Bisect Segments
Day 10-----G.CO.12 &
G.CO.13 Putting it to
Day 5 , 6, & 7 -G.CO.12
Copy Angles and
Bisect Angles
G.CO.13
• G-CO.13. Construct an equilateral
triangle, a square, and a regular
hexagon inscribed in a circle.
GEOMETRY
• Geo: earth
• Metres: measure
• Geometry began as the study of
earth measure.
GEOMETRIC CONSTRUCTIONS
• "Construction" in Geometry means to draw
shapes, angles or lines accurately.
• These constructions use only compass,
straightedge (i.e. ruler) and a pencil.
• This is the "pure" form of geometric
construction - no numbers involved!
Inscribed In:
• Inscribed Circle
• Inscribed Polygon
• A circle is inscribed
in a polygon if the
sides of the polygon
are tangent to the
circle.
• A polygon is
inscribed in a circle if
the vertices of the
polygon are on the
circle.
Equilateral Triangle:
• An equilateral triangle is a
triangle whose sides are all
congruent
REGULAR HEXAGON:
• A hexagon (six sided polygon)
with congruent sides and angles
Creating an Inscribed Hexagon
• This is one of the easiest constructions ever.
• The radius of a circle can be struck around a
circle exactly six times.
• Lets watch:
http://www.mathopenref.com/constinhexagon.html
• Time to Try!!
Using Geometry in Design:
MANDALA
QUESTIONS
Having Kittens
Work out whether this number of
descendants is realistic.
Here are some facts that you will
need:
P-14
HAVING KITTENS
• Can you make a diagram or table to show
what is happening?
• Can you now look systematically at what
happens to her kittens? And their kittens?
• Do you think the first litter of kittens will have
time to grow and have litters of their own?
Then what about their kittens?
• What have you assumed here?
Collaborative Activity
• Work in groups of two.
• Produce a solution that is better than your
individual solution.
• Take turns to explain how your did the task
and how you now thing it could be improved,
then put your individual work aside. Try to
produce a joint solution to the problem.
Assessing Sample Student Responses
Your task is to correct the work and write
comments about its accuracy and organization.
– What has the student done correctly?
– What assumptions has he or she made?
– How could the solution be improved?
P-17
Sample Response: Alice
P-18
Sample Response: Wayne
P-19
Sample Response: Ben
P-20
Reviewing Work
•
I have selected the important facts and used them to solve the
problem.
•
I am aware of the assumptions I have made and the effect these
assumptions have on the result.
•
I have used more than one method
•
I have checked whether my results make sense and improved my
method if need be.
•
I have presented my results in a way that will make sense to
others.
P-21
G.CO.12
• G-CO.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.
TODAY WE WILL
• Copy a segment.
• Bisect a segment.
UNDEFINED TERMS
• Undefined terms are basic terms we need to
describe the shape and size of objects.
• There are three undefined terms in
geometry:
•Point, line, and plane
POINT
• Point (0-D): A location.
• Example: The following is a diagram of points
A, B, C, and Q:
SPACE
• Space: the set of all points.
LINE
• Line (1-D): A series of points that extend in
two directions without end.
• Example: The following is a diagram of two
lines: line AB and line HG.
• The arrows signify that the lines drawn extend
indefinitely in each direction.
PLANE
• Plane (2-D): a flat, two-dimensional object. A
plane must continue infinitely in all directions
and have no thickness at all.
• A plane can be defined by at least three noncollinear points or renamed by a script capital
letter.
GEOMETRIC CONSTRUCTIONS
• "Construction" in Geometry means to draw
shapes, angles or lines accurately.
• These constructions use only compass,
straightedge (i.e. ruler) and a pencil.
• This is the "pure" form of geometric
construction - no numbers involved!
SEGMENT
• Line Segment: part of a line containing two
endpoints and all points between them.
A
AB or BA
B
Getting Started
• In your notes draw two segments. Similar to
the ones below.
COPYING A SEGMENT
• Given: 𝑹𝑺
• Construct: 𝑿𝒀 so that 𝑿𝒀  𝑹𝑺
• http://www.mathopenref.com/constcopyseg
ment.html
PRACTICE #1:
•Given: 𝑹𝑺
•Construct: 𝑊𝑍 = 𝟑𝑹𝑺
PRACTICE #2:
• Given: 𝑹𝑺 & 𝑷𝑸
• Construct: 𝐻𝐺 = 𝑃𝑄 + 2𝑅𝑆.
PRACTICE #3:
• Given: 𝑹𝑺 & 𝑷𝑸
• Construct: 𝐻𝐺 = 3𝑃𝑄 - 2𝑅𝑆.
Bisect
• To divide into two equal
parts.
• You can bisect lines,
angles, and more.
• The dividing line is called
the "bisector
Midpoint
• Midpoint: The point of a line segment that
divides it into two parts of the same length.
• 𝑨𝑪  𝑪𝑬
PERPENDICULAR BISECTORs
 Perpendicular Lines: two lines that intersect
to form right angles. (SYMOBOL )
 Perpendicular Bisector of a Segment: a line,
segment, or ray that is perpendicular to the
segment at its midpoint, thereby bisecting the
segment into two congruent segments.
BISECTING A SEGMENT
 http://www.mathopenref.com/constbisectline.html
Practice #4:
• Given: 𝑨𝑩
• Construct: 𝑿𝒀𝑨𝑩 at the midpoint M of 𝑨𝑩.
Practice #5:
• Given: 𝑨𝑩
• Construct: Divide AB into 4 congruent
segments.
QUESTIONS
G.CO.12
• G-CO.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.
TODAY WE WILL
• Copy an angle.
• Bisect an angle
RAY
• Ray: part of a line that consists of
one endpoint and all points of the
line extending in one direction
B
A
AB , not BA
ANGLES ()
• An angle is two rays meeting at a
common vertex.
ANGLES ()
• Angle: a pair of rays that
share a common endpoint.
• The rays are called the sides of the angle.
• The common endpoint is called the vertex of
the angle.
ANGLES ()
• Obtuse angle
90 < mA < 180
A
• Right angle
m  A = 90
A
• Acute angle
0 < mA < 90
A
COPYING AN ANGLE
• The angle RPQ has the same
measure as BAC
• http://www.mathopenref.com/constcopyangl
e.html
Practice #1
• Given Acute Angle: ABC
• Duplicate ABC, so that ABC = A’B’C’
Practice #2
• Given Obtuse Angle: DEF
• Duplicate DEF, so that DEF = D’E’F’
Practice #3
• Given Acute Angle: ABC
• Construct: XYZ = 2mABC.
Practice #4
• Given: Two acute angles:
MNO & QRS
• Construct MNO + QRS = TUV
Angle Bisector
• A ray that divides that angle into
two congruent coplanar angles.
• Its endpoint is at the angle vertex.
Bisecting an Angle
 Given: A
 Construct: 𝑨𝑿, the bisector of A
 Step 1: Put the compass point on the vertex of A. Draw an
arc that intersects the sides of A. Label the points of
intersection B and C.
 Step 2: Put the compass on point C and draw an arc. With
the same compass setting, draw an arc using point B. Be sure
the arcs intersect. Label the point where the two arcs
intersect as X.
 Step 3: Draw 𝐴𝑋
 𝑨𝑿, the bisector of A
Practice #5
• Draw two angles (one obtuse and one acute)
and bisect each one of them.
QUESTIONS
G.CO.12
• G-CO.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.13
• G-CO.13. Construct an equilateral
triangle, a square, and a regular
hexagon inscribed in a circle.
TODAY WE WILL
• Constructing perpendicular lines,
including the perpendicular
bisector of a line segment
• Construct a square inscribed in a
circle.
Perpendicular Lines
• Lines that intersect at 90
• Symbol: 
Constucting Perpendicular Lines
http://www.mathopenref.com/constperplinepoint.html
Practice #1:
• Construct a line perpendicular to
line l through point Q.
Constructing a Perpendicular to an
external point
http://www.mathopenref.com/constperplinepoint.html
Practice #2:
• Draw a figure like the given one.
Then construct the line through
point P and perpendicular to 𝑅𝑆
Practice #3:
• Draw a figure like the given one.
Then construct the line through
point P and perpendicular to 𝑅𝑆.
•
Inscribed In:
• Inscribed Circle
• Inscribed Polygon
• A circle is inscribed
in a polygon if the
sides of the polygon
are tangent to the
circle.
• A polygon is
inscribed in a circle if
the vertices of the
polygon are on the
circle.
SQUARE
• A square is a parallelogram with
four congruent sides and four
right angles.
SQUARE
• In a square, look at the diagonals.
• What do you notice?
• How can this be used to
construct a square?
SQUARE
A square is a parallelogram with four
congruent sides and four right
angles.
• A few properties of a square:
–1. The diagonals are congruent.
–2. The diagonals are perpendicular.
Creating and Inscribed Square
– Given: Circle O.
– Construct: Square ABCD.
–
–
–
–
–
–
Step 1: Label any point on the circle A.
Step 2: Construct line AO.
Step 3: Label the second intersection of line AO with the circle point C.
Step 4: Construct segment AC.
Step 5: Construct a  to AC through point O.
Step 6: Label the intersections of the perpendicular with the circle points
D and B.
– Step 7: Construct segment DB.
– Step 8: Construct segments AB, BC, CD, and DA.
– ABCD is the required square.
QUESTIONS