Section 1: Introduction to Geometry · PDF fileSection 1: Introduction to Geometry The following Mathematics Florida Standards will be covered in this section: MAFS.912.G-CO.1.1 Know

Section 1: Introduction to Geometry The following Mathematics Florida Standards will be covered in this section: MAFS.912.G-CO.1.1 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.

MAFS.912.G-CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

MAFS.912.G-CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MAFS.912.G-CO.1.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure. Specify a sequence of transformations that will carry a given figure onto itself.

Section 1: Introduction to Geometry 1

MAFS.912.G-CO.4.12 Make formal geometric constructions with a variety of tools and methods. 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.

MAFS.912.G-GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

MAFS.912.G-GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Videos in this Section

Video 1: Basics of Geometry Video 2: Midpoint and Distance in the Coordinate Plane –

Part 1 Video 3: Midpoint and Distance in the Coordinate Plane –

Part 2 Video 4: Partitioning a Line Segment Video 5: Parallel and Perpendicular Lines Video 6: Introduction to Transformations Video 7: Examining and Using Transformations – Part 1 Video 8: Examining and Using Transformations – Part 2 Video 9: Basic Constructions


Section 1 – Video 1 Basics of Geometry

What is geometry? Geometry means “___________ __________________,” and it is concerned with the properties of points, lines, planes and figures. What concepts do you think belong in this branch of mathematics? Why does geometry matter?

When is geometry used in the real world?


Points, lines, and planes are the building blocks of geometry. Consider the following definitions. Draw a representation for each one and fill in the appropriate notation on the chart below.

Definition Representation Notation A point is a precise location or place on a plane. It is usually represented by a dot.

A line is a straight path that continues in both directions forever. Lines are one-dimensional.

A line segment is a portion of a line lying between two points.

A ray is piece of a line that starts at one point and extends infinitely in one direction.

A plane is a flat two-dimensional object. It has no thickness and extends forever.


Definition Representation Notation An angle is formed by two rays with the same endpoint.

The point where the rays meet is called the vertex.

Parallel lines are two lines on the same plane that do not intersect.

Perpendicular lines are two intersecting lines that form a 90° angle.

What can you say about multiple points on a line segment?

Segment Addition Postulate If three points, !,!, and !, are collinear and ! is between ! and !, then !" + !" = !".

!"#$%&'!$()*+,-./,0+%1%!20*304+


Try It!

Consider the diagram below.

The following geometry figures are represented in the diagram. For each figure, give at most 3 names that represents that figure in the diagram above. Figure Name(s) represented in the figure Point Line Line Segment Plane Ray Angle Parallel Lines Perpendicular Lines

Segment Addition Postulate

! ! ! !

! ℳ

! !

!"#$ ! !"#$ !

Use the word bank to complete the definitions below. Draw a representation of each one.

Word Bank

Parallel Planes ♦ Coplanar ♦ Parallel Lines

Collinear ♦ Nonlinear

Definition

Points that lie on the same plane are ______________________.

Points that lie on the same line are ______________________.

Drawing

Let’s Practice! Consider the figure below.

Select all the statements that apply to this figure. □ Points !, !, !, and ! are coplanar in ℛ. □ Points !, !, !, and ! are collinear. □ Points !, !, and ! are collinear and coplanar in ℛ. □ Point ! lies on !". □ Points !,! and ! are coplanar in ℛ. □ Points !, !, ! and ! lie on plane ℛ. □ !" + !" = !"

Try It! Plane ! contains !" and !", and it also intersects !" only at point !. Sketch plane !.

For points, lines, and planes, you need to know certain postulates.

Let’s Practice! Let’s examine the following postulates A through F.

A. Through any two points there is exactly one line. B. Through any three non-collinear points there is exactly

one plane. C. If two points lie in a plane, then the line containing those

points will also lie in the plane. D. If two lines intersect, then they intersect in exactly one

point. E. If two planes intersect, then they intersect in exactly one

line. F. Given a point on a plane, there is one and only one line

perpendicular to the plane through that point.

A postulate is a statement that we take to be automatically true. We do not need to prove that a postulate is true because it is something we assume to be true.

Use postulates A through F to match each postulate with its visual representation.

!

!

!

!

!

ℳ

!

!"#$ !

!

!

!

!

!

! !"#$ !

!

!

!

! !

BEAT THE TEST!

1. Consider the following figure.

Select all the statements that apply to this figure.

o Line ! is perpendicular through point ! to plane !. o Points !, !, !, and ! are coplanar in !. o Points !, !, and ! are collinear. o !" is longer than !". o !" and !" are coplanar in !.

!"

#

$

%

line &

'

!"#$ !

Section 1 – Video 2

Midpoint and Distance in the Coordinate Plane – Part 1

Consider the line segment displayed below.

The length of is _____ centimeters.

____________________ is an amount of space (in certain

units) between two points in a ______________.

Draw a point halfway between point and point . Label this

point .

What is the length of ?

What is the length of ?

Point is called the _________________ of .

Why do you think it’s called the midpoint?

��

Let’s Practice!

Consider with midpoint �.

What can be said of � and � ?

If the length of � is represented by � + and the length of � is . What is the value of �?

�

Consider the line segment below.

If the length of is 128 cm, then what is �? What is the length of �? What is the length of � ? Is point � the midpoint of ? Justify your answer.

�

� + �� − ��

Try It!

Diego lives in Gainesville and Anya lives in Jacksonville. Their

houses are miles apart.

Diego argues that in a straight line distance, Middleburg is

halfway from his house and Anya’s house. Is Diego right?

Justify your reasoning.

Midpoint and distance can also be calculated on a

coordinate plane.

The coordinate plane is a plane that is divided into ________

regions by a horizontal line (______________) and a vertical line

(______________).

The location, or coordinates, of a point are given by an

ordered pair (__________).

Consider the following graph.

Name the ordered pair that represents point .

Name the ordered pair that represents point .

How can we find the midpoint of this line?

The midpoint of is (___ , ____).

Let’s consider points and on the coordinate plane below.

Write a formula that can be used to find the midpoint of any

two given points.

� ,

� ,

Let’s Practice!

has coordinates , . has coordinates − , . Find the

midpoint of .

Try It!

Consider the line segment in the graph below.

Find the midpoint of .

Let’s Practice! � is the midpoint of . has coordinates − ,− and � has

coordinates , . Find the coordinates of .

Try It!


Café 103 is equidistant from Metrics School and Angles Lab.

All three locations are collinear. The Metrics School is located

at point , on a coordinate plane, and Café 103 is at point , . Find the coordinates of Angles Lab.

Metrics School

Café 103

Section 1 - Video 3 Midpoint and Distance in the Coordinate Plane – Part 2 Consider !"#below.

Draw point $ on the above graph at 2, 2 .

What is the length of !$?

What is the length of "$?

Triangle !"$ is a right triangle. Use the Pythagorean theorem to find the length of !".

!#"#

Let’s consider the figure below.

Write a formula to determine the distance of any line segment.

(!(*+, ,+)#

.#(*/, ,/)#

Let’s Practice! Find the distance of 01.

0#

1#

Try It! Consider triangle !"$ graphed on the coordinate plane.

Use the distance formula to find lengths of !", "$ and !$. Round to the nearest tenth.

!!

"!

$!

BEAT THE TEST!

1.!Consider the following figure.

Which of the following statements are true? Select all that apply.

□ The midpoint of !2 has coordinates −4

/ ,5/ .

□ 60 is exactly 5 units. □ !6 is exactly 3 units. □ 12 is longer than 01. □ The perimeter of quadrilateral !"$6 is about 16.6 units. □ The perimeter of quadrilateral !602 is about 18.8 units. □ The perimeter of triangle 012 is 9 units.

A

B C

E

G

F

D


Partitioning a Line Segment

What do you think it means to partition?

How can a line segment be partitioned?

In the previous section, we worked with the ,

which partitions a segment into a : ratio.

Why does the midpoint partition a segment into a : ratio?

How can be divided into a : ratio?

A ratio compares two numbers.

A : ratio is stated as (or can also be written

as) “1 to 1”.

Consider the following line segment where point partitions

the segment into a : ratio.

How many sections are in between points and ?



In relation to , how long is ?

In relation to , how long is ?

Let’s call these ratios �, a fraction that represent a part to a

whole.

When partitioning a directed line segment into two segments,

when would your ratio � be the same value for each

segment? When would it be different?

The following formula can be used to find the coordinates of

a given point that partitions a line segment into ratio �.

, = ( + � − , + � − )

Let’s Practice!

If asked to find the coordinates of a point that partitions a

segment into a ratio of : , what is the value of �?

What if you want to partition the segment into a ratio of : ?

What is the value of �?

has coordinates , . has coordinates , . Find the

coordinates of point that partition in the ratio : .

What if one of the parts of a ratio is actually the whole line,

instead of a ratio of two smaller parts or segments?

Let’s Practice!

Points , , and are collinear and : = . Point is

located at the origin, point is located at , , and point is

located at − , .

What are the values of and ?

Try It!


Find the coordinates of point that partition in the ratio : .

in the coordinate plane has endpoints with coordinates − , and 8, − . Graph and find two possible locations

for point so that divides into two parts with lengths in a

ratio of 1:3.

BEAT THE TEST!

1. Consider the directed line segment from − , to � , .

Points , , and are on � .

Part A: Use the points to complete the statements below.

The point _______ partitions � in a : ratio.



The ratio : � = ___________.

Part B: Draw � and points , , and . Identify the partition

of � in a : ratio.

− , ( , ) ,


Parallel and Perpendicular Lines

Graph A Graph B

These lines are ____________. These lines are ______________.

The symbol used to indicate The symbol used to indicate

parallel lines is _____. perpendicular lines is _____.

Choose two points on each graph and use the slope formula, −− , to verify (prove) your answer.

� �

�

�

What do you notice about the slopes of the parallel lines?

What do you notice about the slopes of the perpendicular

lines?

What happens if the lines are not shown on a graph, but

rather in an equation?

Let’s Practice!

Indicate whether the lines parallel, perpendicular, or neither.

Justify your answer.

= and = +

− = and + =

+ = and − =

= and = −

Try It!

Match each of following with the equations below. Write the

letter of the appropriate equation in the column beside each

item.

A. = − B. = − + C. − = − D. − = −

A line parallel to = +

A line perpendicular to =

A line perpendicular to + =

A line parallel to + =

Let’s Practice!

Write the equation of the line passing through − , and

perpendicular to + = .

Try It!

Suppose the equation for line � is given by = − − .

If line � and line are perpendicular and the point − , lies

on line , then write an equation for line B.

Consider the graph below.

Name a set of lines that are parallel. Justify your answer.

Name a set of lines that are perpendicular? Justify your

answer.

� �

�

BEAT THE TEST!

1. The equation for line � is given by = − − . Suppose line � is parallel to line , and line � is perpendicular to line �.

Point , lies on both line and line �.

Part A: Write an equation for line .

Part B: Write an equation for line �.

2. A parallelogram is a four-sided figure whose opposite sides

are parallel and equal in length. Alex is drawing

parallelogram � on a coordinate plane. The

parallelogram has the coordinates � , , , − , and , − .

Which of the following coordinates should Alex use for the

point ?

A , −

B , −

C , −

D ,


Introduction to Transformations

What do you think happens when you transform a figure?

What are some different ways that you can transform a

figure?

In geometry, transformations refer to the ______________ of

objects on a coordinate plane.

A pre-figure or pre-image is the original object.

We use the prime notation [ ′ ] to represent a transformed

figure of the original figure.

For example, if is a point on or vertex of the original

figure, then ′ is one transformation of A.

Consider the following geometric transformations.

Dilation Reflection Rotation Translation

Match each transformation with its corresponding graphic

representation below.

’ ’

’

’ ’ ’

’ ’

’

’ ’ ’

There are two main categories of transformations: rigid and

non-rigid.

Which kind of transformation changes the shape and/or size

of the pre-image?

Which kind of transformation does not change the size or the

shape of the pre-image?

There are four common types of transformations:

A rotation turns the shape around a center point.

A translation slides the shape in any direction.

A dilation changes the size of an object through an

enlargement or a reduction.

A reflection flips the object over a line (as in a mirror

image).

In the table below, indicate whether the transformation is

rigid or non-rigid and justify your answer.

Transformation Rigid/Non-Rigid Justification

Translation o Rigid

o Non-Rigid

Reflection o Rigid

o Non-Rigid

Rotation o Rigid

o Non-Rigid

Dilation o Rigid

o Non-Rigid

Let’s Practice!

Consider in the coordinate plane below.

Write the coordinates of each endpoint, the length of the

segment, and the midpoint of the segment.

______, ______

______, ______

Length: ______ units

Midpoint: ______, ______

Write the coordinates of ′ and ′ after the following

transformations.

Transformations Coordinates AB is translated units to the left. AB is dilated by a factor of (centered at the

origin). AB is rotated ° clockwise about the origin. AB is reflected over the − axis?

Try It!

Consider the transformations of in the previous problem.

Trace the lines and identify the transformations on the graph.

What are the ′ and ’ coordinates in each transformation

below? What are the length and midpoint of each segment

indicated in the chart?

Transformation Coordinates Length Midpoint

Translation ′ ____, ____ ; ′ ____, ____

Dilation ′ ____, ____ ; ′ ____, ____

Rotation ′ ____, ____ ; ′ ____, ____

Reflection ′ ____, ____ ; ′ ____, ____

’ ’

’ ’ ’

’ ’ ’

BEAT THE TEST!

1. Three rays share the same vertex , as shown in the

coordinate plane below.

Part A: Which figure represents a reflection across the

-axis?

A Figure A

B Figure B

C Figure C

D None

Part B: Which of the following statements are true about the

figure? Select all that apply.

A rotation of ° will carry the object onto itself.

A reflection of the figure along the -axis carries

the figure to Quadrant II.

In Figure A, ′, ′ = + , . If the vertex of Figure A is translated + , − 9 ,

it will carry onto the vertex of Figure B.

Figure C is a reflection on the -axis of Figure A.

Figure A

Figure B Figure C


Examining and Using Transformations – Part 1

Let’s further examine transformations.

Translation

A translation "slides" an object a fixed distance in a given

direction, preserving the same ___________ and __________.

Suppose a geometric figure is translated ℎ units along the

-axis and � units along the -axis. We use the following

notation to represent the transformation:

�ℎ,� , = + ℎ, + �

� ,− , translates the point , units ______ and

units _______.

What is the transformation that translates the point ,

units to the left and units up?

Let’s Practice!

Translate triangle according to the transformation � ,− , = + , − . Write the coordinates for triangle ′ ′ ′.

′ ______, ______

′ ______, ______

′ ______, ______

Try It!

undergoes the translation �ℎ,� , = + ℎ, + � , such

that ′ , and ′ , . What are the values of ℎ and �?

ℎ = ___________ units

� = ___________ units

Dilation

Dilation is a transformation that produces an image that is the

same ______________ as the original, but is a different

________________.

The function � , = � , � changes the shape of the

figure by a factor of �. Here, � represents the scale factor.

The scale factor refers to how much the figure grows or

shrinks.

, enlarges the shape of the figure by a factor of

______.

What would 13 , do to the shape of the figure?

Let’s Practice! has coordinates − , and , − . has

coordinates , − and , .

Find the coordinates of ′ ′ after a dilation with a scale factor

of .

Find the coordinates of ′ ′ after a dilation with a scale factor

of 4.

Try It!

What is the scale factor for the dilation of into ′ ′ ′?

has coordinates − , − and − , − . Find the

coordinates of ′ ′ after a dilation with a scale factor of .

’ ’ ’


Examining and Using Transformations – Part 2

Rotation

A rotation changes the __________ of a figure by moving it

around a fixed point to the right (clockwise) or to the left

(counterclockwise).

The function , rotates the point , �° counterclockwise around the origin.

, rotates the point , ° counterclockwise.

− , rotates the point , �° clockwise around the

origin.

− , rotates the point , ° clockwise.

Let’s consider the following graph.

Use the graph to help you determine the coordinates for ’, ’ .

Counterclockwise Clockwise ° Rotation: ° , = ________ − ° , = ________ ° Rotation: ° , = ________ − ° , = ________ ° Rotation: ° , = ________ − ° , = ________

Make generalizations to complete the following.

Counterclockwise Clockwise ° Rotation: ° , = ________ − ° , = ________ ° Rotation: ° , = ________ − ° , = ________ ° Rotation: ° , = ________ − ° , = ________

What happens if the rotation is ° clockwise or

counterclockwise?

Let’s Practice! � has endpoints , and � , . Rotate � clockwise by

90 degrees about the origin. What are the endpoints of the

new line segment?

Try It!

Consider the following figure.

Part A: Rotate figure ° counterclockwise about the

origin. Graph the new figure on the coordinate plane,

and complete each blank below with the appropriate

coordinates.

° ____, ____ = ′ ____, ____ ° ____, ____ = ′ ____, ____ ° ____, ____ = ′ ____, ____

Part B: Rotate figure ° clockwise about the origin.

Graph the new figure on the coordinate plane, and

complete each blank below with the appropriate

coordinates.

− ° ____, ____ = ′ ____, ____ − ° ____, ____ = ′ ____, ____ − ° ____, ____ = ′ ____, ____

Reflection

A reflection is a mirrored version of an object. The image does

not change __________ but the figure itself reverses.

The function �� , reflects the point , over the given

line. For instance, � −� � , reflects the point , over the

-axis.

Let’s examine the line reflections of the point , over the

-axis, -axis, = , and = − .

Reflection over Notation New coordinates

-axis � −� � ,

-axis � −� � , = � = , = − � =− ,

Make generalizations to complete the following.

Reflection over Notation New coordinates

-axis � −� � ,

-axis � −� � , = � = , = − � =− ,

Let’s Practice!

Suppose the line segment whose endpoints are , and , is reflected over the -axis. What are the coordinates ’ and ’?

Suppose the line segments whose endpoints are − , and − , − is reflected over = . What are the coordinates ′ and ′?

Try It!

Suppose the line segment whose endpoints are − , −

and − , is reflected over = − . What are the

coordinates of ′ ____, ____ and ′ ____, ____ ?

Graph ′ ′ on the coordinate plane below.

BEAT THE TEST!

1. − , − , − , − and − , form figure .

Part A: Gladys transformed figure into ′ ′ ′. Which of

the following represents her transformation?

A , = ,

B � ,5 , = + , +

C � = , = ,

D � =− , = − , −

Part B: She then transformed ′ ′ ′ into ′′ ′′ ′′. What is the

transformation?

, → ____________________, ____________________

’ ’’

’ ’’

’ ’’


Basic Constructions

What do you think the term geometric constructions implies?

The following tools are used in geometric constructions.

Straightedge Compass

Which of the tools would help you draw a line segment?

Which of the tools would help you draw a circle?

Constructions also involves labeling points where lines or arcs

intersect.

An arc is a section of the ________________ of a circle, or any

curve.

Consider the following figure where was constructed

perpendicular to .

Where was a straight edge used? Label each part of the

figure that evidences use of a straightedge with the letters SE.

Where was a compass used? Label each part of the figure

that evidences use of a compass with the letter C.

Let’s Practice!

Follow the instructions below for copying .

1. Mark a point that will be one endpoint of the new line

segment.

2. Set the point of the compass on point of the line segment

to be copied.

3. Adjust the width of the compass to point . The width of the

compass width is now equal to the length of .

4. Without changing the width of the compass, place the

compass point on . Keeping the same compass's width,

draw an arc approximately where the other endpoint will

be created.

5. Pick a point on the arc that will be the other endpoint of

the new line segment.

6. Use the straightedge to draw a line segment from to .

Try It!

Construct a copy of the from the triangle in such a way

that the copy has one endpoint at .

Write down the steps you followed for your construction.

Consider the following figure where is the perpendicular

bisector of .

Make a conjecture as to why is called the perpendicular

bisector of .

A bisector divides lines, angles, and shapes into two equal

parts.

A perpendicular bisector is a perpendicular line that cross a

line segment (at 90°) into two equal parts.

When you make a conjecture, you make an educated guess

based on what you know or observe.

Let’s Practice!

Follow the instructions below for constructing the

perpendicular bisector of .

1. Start with .

2. Place your compass point on , and stretch the compass

more than halfway to point .

3. Draw large arcs both above and below the midpoint of .

4. Without changing the width of the compass, place the

compass point on . Draw two arcs so that they intersect

the arcs you drew in step 3.

5. With your straightedge, connect the two points of where

the arcs intersect.

Construct a perpendicular bisector of line segment .

Try It!

Construct the perpendicular bisector of and .

What type of geometric construction is formed when you

draw the diagram below?

What is the next logical step(s) to complete the construction

above?

R

P Q

BEAT THE TEST!

1. Which of the following best describes the construction?

A ∥ .

B ≅ .

C is the midpoint of �.

D is the midpoint of �.

�

2. Fernando was constructing a perpendicular line at a

point on the line below.

The figure below represents a depiction of the partial

construction Fernando made.

What should be the next logical step to his geometric

construction?

A Increase the compass to almost double the width to

create another line.

B From , draw an arc below .

C Without changing the width of the compass, repeat

the drawing process from point , making the two

arcs cross each other at a new point called .

D Close the compass and use the straight edge to draw

a line from the midpoint of the arc to point .

Documents

Section 1: Introduction to Geometry · PDF fileSection 1: Introduction to Geometry The following Mathematics Florida Standards will be covered in this section: MAFS.912.G-CO.1.1 Know