GEOMETRY

2013-2014

Geometry Tools

August 27, 2013• Toolbox

• Automaticity

STANDARDSTrimester 1

Day 1-----G.CO.13 Equilateral Triangle & Regular Hexagon

Day 2 & 3 -----FAL: Having Kittens

Day 3 & 4---G.CO.12Copy Segments & Bisect Segments

Day 5 , 6, & 7 --G.CO.12Copy Angles and Bisect Angles

Day 8 -----G.CO.12 Perpendiculars and G.CO.13 Inscribed Square

Day 9-----G.CO.12 Parallel Lines

Day 10-----G.CO.12 & G.CO.13 Putting it to

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• A circle is inscribed

in a polygon if the sides of the polygon are tangent to the circle.

• Inscribed Polygon• 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

P-14

Work out whether this number of descendants is realistic.Here are some facts that you will need:

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

P-21

• 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.

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 non-collinear 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 B

AB or BA

Getting Started• In your notes draw two segments. Similar to

the ones below.

COPYING A SEGMENT

• Given: • Construct: so that

• http://www.mathopenref.com/constcopysegment.html

PRACTICE #1:

•Given: •Construct: =

PRACTICE #2:

• Given: & • Construct: = + 2.

PRACTICE #3:

• Given: & • Construct: = 3- 2.

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.•

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: • Construct: at the midpoint M of .

Practice #5:• Given: • 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

A

B

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 < mA < 180

• Right angle m A = 90

• Acute angle 0 < mA < 90

A

A

A

COPYING AN ANGLE

• The angle RPQ has the same measure as BAC

• http://www.mathopenref.com/constcopyangle.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 = 2mABC.

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: , 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 , 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 Lineshttp://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

Practice #3:

• Draw a figure like the given one. Then construct the line through point P and perpendicular to

•

Inscribed In: • Inscribed Circle• A circle is inscribed

in a polygon if the sides of the polygon are tangent to the circle.

• Inscribed Polygon• 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