WIXLIB

Section 1: Introduction to GeometryThe following Mathematics Florida Standards will be coveredin this .G-CO.1.4MAFS.912.G-CO.1.5Know precise definitions of angle,circle, perpendicular line, parallelline, and line segment, based on theundefined notions of point, line,distance along a line, and distancearound a circular arc.Represent transformations in theplane using, e.g., transparencies andgeometry software; describetransformations as functions that takepoints in the plane as inputs and giveother points as outputs. Comparetransformations that preservedistance and angle to those that donot.Develop definitions of rotations,reflections, and translations in terms ofangles, circles, perpendicular lines,parallel lines, and line segments.Given a geometric figure and arotation, reflection, or translation,draw the transformed figure. Specifya sequence of transformations thatwill carry a given figure onto itself.Section 1: Introduction to Geometry1

.2.6Make formal geometric constructionswith a variety of tools and methods.Copying a segment; copying anangle; bisecting a segment; bisectingan angle; constructing perpendicularlines, including the perpendicularbisector of a line segment; andconstructing a line parallel to a givenline through a point not on the line.Prove the slope criteria for paralleland perpendicular lines and usethem to solve geometric problems.Find the point on a directed linesegment between two given pointsthat partitions the segment in a givenratio.Videos in this SectionVideo 1:Video 2:Video 3:Video 4:Video 5:Video 6:Video 7:Video 8:Video 9:Basics of GeometryMidpoint and Distance in the Coordinate Plane –Part 1Midpoint and Distance in the Coordinate Plane –Part 2Partitioning a Line SegmentParallel and Perpendicular LinesIntroduction to TransformationsExamining and Using Transformations – Part 1Examining and Using Transformations – Part 2Basic ConstructionsSection 1: Introduction to Geometry2

Section 1 – Video 1Basics of GeometryWhat is geometry?Geometry means “ ,” and it isconcerned with the properties of points, lines, planes andfigures.What concepts do you think belong in this branch ofmathematics?Why does geometry matter?When is geometry used in the real world?Section 1: Introduction to Geometry3

Points, lines, and planes are the building blocks of geometry.Consider the following definitions. Draw a representation foreach one and fill in the appropriate notation on the chartbelow.DefinitionA point is a preciselocation or place on aplane. It is usuallyrepresented by a dot.RepresentationNotationA line is a straight paththat continues in bothdirections forever. Linesare one-dimensional.A line segment is aportion of a line lyingbetween two points.A ray is piece of a linethat starts at one pointand extends infinitely inone direction.A plane is a flattwo-dimensional object. Ithas no thickness andextends forever.Section 1: Introduction to Geometry4

DefinitionAn angle is formed bytwo rays with the sameendpoint.RepresentationNotationThe point where the raysmeet is called the vertex.Parallel lines are twolines on the same planethat do not intersect.Perpendicular lines aretwo intersecting lines thatform a 90 angle.What can you say about multiple points on a line segment?!"# %&'! ()* ,-./,0 %1%!20*304 Segment Addition PostulateIf three points, !, !,  and !, are collinear and ! isbetween ! and  !, then !" !" !".Section 1: Introduction to Geometry5

Try It!Consider the diagram below.!!!!!!!"# !ℳ!!"#  !The following geometry figures are represented in thediagram. For each figure, give at most 3 names thatrepresents that figure in the diagram above.FigurePointLineLine SegmentPlaneRayAngleParallel LinesPerpendicularLinesSegment AdditionPostulateName(s) represented in the figure

Use the word bank to complete the definitions below. Draw arepresentation of each one.Word BankParallel Planes Coplanar Parallel LinesCollinear NonlinearDefinitionPoints that lie on thePoints that lie on thesame plane aresame 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 atpoint !.Sketch plane !.

For points, lines, and planes, you need to know certainpostulates.A postulate is a statement that we take to beautomatically true. We do not need to provethat a postulate is true because it issomething we assume to be true.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 exactlyone plane.C. If two points lie in a plane, then the line containing thosepoints will also lie in the plane.D. If two lines intersect, then they intersect in exactly onepoint.E. If two planes intersect, then they intersect in exactly oneline.F. Given a point on a plane, there is one and only one lineperpendicular to the plane through that point.

Use postulates A through F to match each postulate with itsvisual representation.!!!!!"# !!!!!!!!"#  !!!!!!!!ℳ

BEAT THE TEST!1. Consider the following figure.line &!"#  !'" !%#Select all the statements that apply to this figure.oooooLine ! is perpendicular through point ! to plane !.Points !, !, !, and ! are coplanar in !.Points !,  !, and ! are collinear.!" is longer than !".!" and !" are coplanar in !.

Section 1 – Video 2Midpoint and Distance in the Coordinate Plane – Part 1Consider the line segment displayed below. The length of ̅̅̅̅ is centimeters. is an amount of space (in certainunits) between two points in a .Draw a point halfway between pointpoint .and point . Label thisWhat is the length of ̅̅̅̅ ?What is the length of ̅̅̅̅?Pointis 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 ̅ ̅̅̅ is . What is the value of ?and the length 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. Theirhouses aremiles apart.Diego argues that in a straight line distance, Middleburg ishalfway from his house and Anya’s house. Is Diego right?Justify your reasoning.

Midpoint and distance can also be calculated on acoordinate plane.The coordinate plane is a plane that is divided intoregions by a horizontal line ( ) and a vertical line( ). The location, or coordinates, of a point are given by anordered 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 pointsandon the coordinate plane below. , ,Write a formula that can be used to find the midpoint of anytwo given points.

Let’s Practice!has coordinatesmidpoint of ̅̅̅̅.,.has coordinates Try It!Consider the line segment in the graph below.Find the midpoint of ̅̅̅̅.,. Find the

Let’s Practice! is the midpoint of ̅̅̅̅ . has coordinates , coordinates , . Find the coordinates of .and hasTry It!Consider the line segment in the graph below.Café 103 is equidistant from Metrics School and Angles Lab.All three locations are collinear. The Metrics School is locatedat point , on a coordinate plane, and Café 103 is at point, . Find the coordinates of Angles Lab.Metrics SchoolCafé 103

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

Let’s consider the figure below.#(*/ , ,/ )#(!(* , , )#Write a formula to determine the distance of any linesegment.

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.FGEADBCWhich of the following statements are true? Select all thatapply. 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.

Section 1 – Video 4Partitioning a Line SegmentWhat do you think it means to partition?How can a line segment be partitioned?In the previous section, we worked with thewhich partitions a segment into a : ratio.,A ratio compares two numbers.A : ratio is stated as (or can also be writtenas) “1 to 1”.Why does the midpoint partition a segment into a :How can ̅̅̅̅ be divided into a :ratio?ratio?

Consider the following line segment where pointthe segment into a : ratio.partitionsHow many sections are in between pointsand ?How many sections are in between pointsand ?How many sections are in between pointsand ?In relation to ̅̅̅̅, how long is ̅̅̅̅?In relation to ̅̅̅̅, how long is ̅̅̅̅?Let’s call these ratios , a fraction that represent a part to awhole.When partitioning a directed line segment into two segments,when would your ratio be the same value for eachsegment? When would it be different?

The following formula can be used to find the coordinates ofa given point that partitions a line segment into ratio ., ( , )Let’s Practice!If asked to find the coordinates of a point that partitions asegment 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 thecoordinates 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 , , andare collinear andlocated at the origin, pointlocated at , .What are the values of:is located atand ? . Point,is, and pointis

Try It!Consider the line segment in the graph below.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 locationsfor point so that divides ̅̅̅ into two parts with lengths in aratio of 1:3.

BEAT THE TEST!1. Consider the directed line segment fromPoints , , and are on ̅̅̅̅ . , ,( , )to ,.,Part A: Use the points to complete the statements below.The point partitions ̅̅̅̅ in a :ratio.The point partitions ̅̅̅̅ in a :ratio.The point partitions ̅̅̅̅ in a :The ratio: .Part B: Draw ̅̅̅̅ and points ,of ̅̅̅̅ in a : ratio., andratio. Identify the partition

Section 1 – Video 5Parallel and Perpendicular LinesGraph AGraph B These lines are .The symbol used to indicateparallel lines is . These lines are .The symbol used to indicateperpendicular 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 perpendicularlines?What happens if the lines are not shown on a graph, butrather 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 theletter of the appropriate equation in the column beside eachitem.A. B. A line parallel to A line perpendicular toA line perpendicular toA line parallel to C. D.