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1.1: Geometry Terms

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Define and use terms, including points, lines, planes, space, and postulates.

Geometric Definitions

A point is an exact location in space . It describes a location, but has no size. Examples are shown below:

A line is infinitely many points that extend forever in both directions. Lines have direction and location and are always straight .

Figure \(\ PageIndex{2}\)

A plane is a flat surface that contains infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper with no thickness that goes on forever.

We can use point , line , and plane to define new terms.

Space is the set of all points extending in three dimensions . Think back to the plane. It extended in two dimensions, what we think of as up/down and left/right. If we add a third dimension, one that is perpendicular to the other two, we arrive at three-dimensional space.

Points that lie on the same line are collinear . \(P\),\(Q\),\(R\),\(S\), and \(T\) are collinear because they are all on line \(w\). If a point U were located above or below line w, it would be non-collinear .

Points and/or lines within the same plane are coplanar . Lines h and i, and points \(A\),\(B\),\(C\),\(D\),\(G\), and \(K\) are coplanar in Plane \(J\). Line \(\overleftrightarrow{KF}\) and point \(E\) are non-coplanar with Plane \(J\).

An endpoint is a point at the end of a line segment . A line segment is a portion of a line with two endpoints. Or, it is a finite part of a line that stops at both ends. Line segments are labeled by their endpoints. Order does not matter.

A ray is a part of a line. It begins with an endpoint and extends forever away from the endpoint in one direction, perfectly straight. A ray is labeled by its endpoint and one other point on the ray. For rays, order does matter. When labeling, put the endpoint under the side WITHOUT the arrow.

An intersection is a point or set of points where lines, planes, segments, or rays overlap.

A postulate is a basic rule of geometry. Postulates are assumed to be true (rather than proven), much like definitions. The following is a list of some basic postulates.

Postulate #1: Given any two distinct points, there is exactly one (straight) line containing those two points.

Postulate #2: Given any three non-collinear points, there is exactly one plane containing those three points.

Postulate #3: If a line and a plane share two points, then the entire line lies within the plane.

Postulate #4: If two distinct lines intersect, the intersection will be one point.

Lines \(I\) and \(m\) intersect at point \(A\).

Postulate #5: If two distinct planes intersect, the intersection will be a line.

Figure \(\PageIndex{13}\)

When making geometric drawings, be sure to be clear and label all points and lines.

What if you were given a picture of a figure or object, like a map with cities and roads marked on it? How could you explain that picture geometrically?

Example \(\PageIndex{1}\)

What best describes San Diego, California on a globe: point, line, or plane?

A city is usually labeled with a dot, or point, on a globe.

Example \(\PageIndex{2}\)

Use the picture below to answer these questions.

Figure \(\ PageIndex {14}\)

• List another way to label Plane \(J\).
• List another way to label line \(h\).
• Are \(K\) and \(F\) collinear?
• Are \(E\),\(B\) and \(F\) coplanar?
• Plane \(BDG\) is one possibility. Any combination of three coplanar points that are not collinear would be correct.
• \(\overleftrightarrow{AB}\). Any combination of two of the letters \(A\), \(B\), or \(C\) would also be correct.
• Yes, they both lie on \(\overleftrightarrow{KF}\).
• Yes, even though \(E\) is not in Plane \(J\), any three points make a distinct plane. Therefore, the three points create Plane \(EBF\).

Example \(\PageIndex{3}\)

What best describes a straight road that begins in one city and stops in a second city: ray, line, segment, or plane?

The straight road connects two cities, which are like endpoints. The best term is segment.

Example \(\PageIndex{4}\)

Answer the following questions about the picture.

• Is line \(l\) coplanar with Plane \(V\), Plane \(W\), both, or neither?
• Are \(R\) and \(Q\) collinear?
• What point belongs to neither Plane V nor Plane \(W\)?
• List three points in Plane \(W\).
• Any combination of \(P\), \(O\), \(T\), and \(Q\) would work.

Example \(\PageIndex{5}\)

Draw and label a figure matching the following description: Line \(\overleftrightarrow{AB}\) and ray \(\overrightarrow{CD}\) intersect at point \(C)\. Then, redraw so that the figure looks different but is still true to the description.

Neither the position of A or B on the line, nor the direction that \(\overrightarrow{CD}\) points matter.

For the second part, this is one way to draw the diagram differently:

For questions 1-5, draw and label a figure to fit the descriptions.

• \(\overrightarrow{CD}\) intersecting \(\overline{AB}\) and Plane \(P\) containing \(\overline{AB}\) but not \(\overrightarrow{CD}\).
• Three collinear points \(A\), \(B\), and \(C\). \(B\) is also collinear with points \(D\) and \(E\).
• \(\overrightarrow{XY}\), \(\overrightarrow{XZ}\), and \(\overrightarrow{XW}\), such that \(\overrightarrow{XY}\) and \(\overrightarrow{XZ}\) are coplanar, but \(\overrightarrow{XW}\) is non-coplanar with both of the other rays.
• Two intersecting planes, \(P\) and \(Q\), with \(\overline{GH}\), where \(G\) is in plane \(P\) and \(H\) is in plane \(Q\).
• Four non-collinear points \(I\), \(J\), \(K\),and \(L\), with line segments connecting all points to each other.
• Name this line in five ways.

Figure \(\PageIndex{18}\)

• Name the geometric figure in three different ways.

Figure \(\PageIndex{19}\)

• Name the geometric figure in two different ways.

Figure \(\PageIndex{20}\)

• What is the best possible geometric model for a soccer field? Explain your answer.
• List two examples of where you see rays in real life.
• What type of geometric object is the intersection of a line and a plane? Draw your answer.
• What is the difference between a postulate and a theorem ?

For 13-16, use geometric notation to explain each picture in as much detail as possible.

Figure \(\PageIndex{23}\)

For 17-25, determine if the following statements are true or false.

• Any two points are collinear.
• Any three points determine a plane.
• A line is two rays with a common endpoint.
• A line segment is infinitely many points between two endpoints.
• A point takes up space.
• A line is one-dimensional.
• Any four points are coplanar.
• \(\overrightarrow{AB}\) could be read “ray \(AB\)” or “ray \(BA\).”
• \(\overleftrightarrow{AB}\) could be read “line \(AB\)” or “line \(BA\).”

Review (Answers)

To see the Review answers, open this PDF file and look for section 1.1.

Additional Resources

Interactive Element

Video: Basic Geometric Definitions Principles - Basic

Activities: Basic Geometric Definitions Discussion Questions

Study Aids: Basics of Geometry Study Guide

Practice: Geometry Terms

Real World: Basic Geometric Definitions

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Creating Definitions (Lesson 1.1)

Unit 1: reasoning in geometry, day 1: creating definitions, day 2: inductive reasoning, day 3: conditional statements, day 4: quiz 1.1 to 1.3, day 5: what is deductive reasoning, day 6: using deductive reasoning, day 7: visual reasoning, day 8: unit 1 review, day 9: unit 1 test, unit 2: building blocks of geometry, day 1: points, lines, segments, and rays, day 2: coordinate connection: midpoint, day 3: naming and classifying angles, day 4: vertical angles and linear pairs, day 5: quiz 2.1 to 2.4, day 6: angles on parallel lines, day 7: coordinate connection: parallel vs. perpendicular, day 8: coordinate connection: parallel vs. perpendicular, day 9: quiz 2.5 to 2.6, day 10: unit 2 review, day 11: unit 2 test, unit 3: congruence transformations, day 1: introduction to transformations, day 2: translations, day 3: reflections, day 4: rotations, day 5: quiz 3.1 to 3.4, day 6: compositions of transformations, day 7: compositions of transformations, day 8: definition of congruence, day 9: coordinate connection: transformations of equations, day 10: quiz 3.5 to 3.7, day 11: unit 3 review, day 12: unit 3 test, unit 4: triangles and proof, day 1: what makes a triangle, day 2: triangle properties, day 3: proving the exterior angle conjecture, day 4: angle side relationships in triangles, day 5: right triangles & pythagorean theorem, day 6: coordinate connection: distance, day 7: review 4.1-4.6, day 8: quiz 4.1to 4.6, day 9: establishing congruent parts in triangles, day 10: triangle congruence shortcuts, day 11: more triangle congruence shortcuts, day 12: more triangle congruence shortcuts, day 13: triangle congruence proofs, day 14: triangle congruence proofs, day 15: quiz 4.7 to 4.10, day 16: unit 4 review, day 17: unit 4 test, unit 5: quadrilaterals and other polygons, day 1: quadrilateral hierarchy, day 2: proving parallelogram properties, day 3: properties of special parallelograms, day 4: coordinate connection: quadrilaterals on the plane, day 5: review 5.1-5.4, day 6: quiz 5.1 to 5.4, day 7: areas of quadrilaterals, day 8: polygon interior and exterior angle sums, day 9: regular polygons and their areas, day 10: quiz 5.5 to 5.7, day 11: unit 5 review, day 12: unit 5 test, unit 6: similarity, day 1: dilations, scale factor, and similarity, day 2: coordinate connection: dilations on the plane, day 3: proving similar figures, day 4: quiz 6.1 to 6.3, day 5: triangle similarity shortcuts, day 6: proportional segments between parallel lines, day 7: area and perimeter of similar figures, day 8: quiz 6.4 to 6.6, day 9: unit 6 review, day 10: unit 6 test, unit 7: special right triangles & trigonometry, day 1: 45˚, 45˚, 90˚ triangles, day 2: 30˚, 60˚, 90˚ triangles, day 3: trigonometric ratios, day 4: using trig ratios to solve for missing sides, day 5: review 7.1-7.4, day 6: quiz 7.1 to 7.4, day 7: inverse trig ratios, day 8: applications of trigonometry, day 9: quiz 7.5 to 7.6, day 10: unit 7 review, day 11: unit 7 test, unit 8: circles, day 1: coordinate connection: equation of a circle, day 2: circle vocabulary, day 3: tangents to circles, day 4: chords and arcs, day 5: perpendicular bisectors of chords, day 6: inscribed angles and quadrilaterals, day 7: review 8.1-8.6, day 8: quiz 8.1 to 8.6, day 9: area and circumference of a circle, day 10: area of a sector, day 11: arc length, day 12: quiz 8.7 to 8.9, day 13: unit 8 review, day 14: unit 8 test, unit 9: surface area and volume, day 1: introducing volume with prisms and cylinders, day 2: surface area and volume of prisms and cylinders, day 3: volume of pyramids and cones, day 4: surface area of pyramids and cones, day 5: review 9.1-9.4, day 6: quiz 9.1 to 9.4, day 7: volume of spheres, day 8: surface area of spheres, day 9: problem solving with volume, day 10: volume of similar solids, day 11: quiz 9.5 to 9.8, day 12: unit 9 review, day 13: unit 9 test, unit 10: statistics and probability, day 1: categorical data and displays, day 2: measures of center for quantitative data, day 3: measures of spread for quantitative data, day 4: quiz review (10.1 to 10.3), day 5: quiz 10.1 to 10.3, day 6: scatterplots and line of best fit, day 7: predictions and residuals, day 8: models for nonlinear data, day 9: quiz review (10.4 to 10.6), day 10: quiz 10.4 to 10.6, day 11: probability models and rules, day 12: probability using two-way tables, day 13: probability using tree diagrams, day 14: quiz review (10.7 to 10.9), day 15: quiz 10.7 to 10.9, day 16: random sampling, day 17: margin of error, day 18: observational studies and experiments, day 19: random sample and random assignment, day 20: quiz review (10.10 to 10.13), day 21: quiz 10.10 to 10.13, learning targets.

Understand the process of writing a good definition requires classifying, differentiating and testing the definition by looking for counterexamples.

Define basic geometric terms by looking at examples and nonexamples.

Activity: What is a Shoe?

Lesson handouts, media locked.

Our Teaching Philosophy:

Experience first, formalize later (effl), experience first.

Geometry Unit 1 Overview and Learning Targets: Word | pdf

While most people agree that what they like about math is that it's black and white, today's opening activity may call that into question. Geometry is full of definitions, but students will realize that definitions are simply decisions (sometimes disputed ones!) about what counts and what does not. By having to write a precise definition of a shoe based on examples and non-examples, we hope students begin to appreciate some of the complexity and perhaps even ambiguity involved with thinking and reasoning mathematically.

To prepare for today's activity, you will need a variety of "shoes" for students to classify. We use pairs of tennis shoes, high heels, flip-flops, socks, slippers, and Barbie shoes, though you may wish to include more or different items. The important point is that the items need to make the definition of shoes somewhat disputable.

Begin by giving students 30-60 seconds to write their own definition of a shoe. Have various students share out their answers. Then take out your bag of "shoes" and hold them up one by one. Students must choose whether to write the item in the "shoe" or "not a shoe" category". Our students tend to be pretty vocal about their opinions and the debate can even get a bit heated! We welcome this as an opportunity to demonstrate that math class this year may be different than what they are used to, and will include plenty of talking as well as generating and critiquing arguments. Allow students to share first in their groups and then with the whole class how they made their decisions.

In question 3, students get to revise their definition so that it clearly distinguishes what counts and what doesn't count as a shoe. In question 4, students critique a dictionary definition of the word shoe.

It’s not about whether students agree which “shoes” fall in what category but that their definition clearly delineates what’s included from what’s excluded. They may or may not agree with the dictionary.com definition, but they should realize that it takes a clear stance on what counts and what doesn't. Good definitions classify and differentiate.

Formalize Later

When we transition from small group work to whole class debrief, we tell students to switch to a red pen. This allows students to clearly see their original thinking and what they learned in the whole class debrief. The margin notes help formalize the learning, adding precise vocabulary, notation, or important connections to the work they've already done during the activity. We encourage students not to erase previous answers, even if they've changed their minds or no longer think their answers are correct. Revising, changing one's mind, and uncertainty are crucial aspects of learning math, and we want to dispel the common notion that all mistakes are to be either avoided or quickly obliterated. We value the thinking process over just the correct answer.

In the debrief we add the words "Examples" and "Non-examples" as markers for what counts and what doesn't. You could even have students re-classify the items based on the dictionary.com definition to emphasize how definitions should take a clear stance on what is included and what is excluded.

We use the "classify" and "differentiate" structure for writing definitions. First, we determine the category of the item to provide a context and then we distinguish the item from similar items in that category.

Students should walk away today knowing that definitions are decisions! A shared understanding of what words mean helps facilitate communication in the math community and beyond. But like with all conventions (especially notation!), there is room for critique and questioning of assumptions. While mathematicians value precision, an overemphasis on proper notation and correct language, especially at the very beginning of a course, can impede students from seeing the bigger picture of the richness and creativity of math, and their place in it. Throughout this course we will strive to give students equitable access to the discipline while promoting their agency within it.

Math Medic Help

1.1: The Right Tool (10 minutes)

CCSS Standards

Building Towards

• HSG-CO.D.12
• HSG-CO.D.13

Routines and Materials

Required Materials

• Geometry toolkits (HS)

The purpose of this warm-up is for students to familiarize themselves with the straightedge and compass.

They will learn to:

• draw a circle
• draw a line segment
• transfer a distance 

In this unit, students start with a small set of tools for construction and editing in a custom applet, called Constructions, which can be found in the Math Tools menu or at ggbm.at/C9acgzUx . These are the GeoGebra tools in that app, those that do the same jobs as a pencil, a compass, and a straightedge.

Three pencil tools:

point plotted on object

point of intersection of objects

Four straightedge tools:

Two compass tools:

circle with center through point

To begin the activity, give students 2 minutes of quiet work time.

Pause the class to:

• demonstrate how to use the Circle tool by creating a circle centered at a given point and passing through another point
• demonstrate how to use the Compass tool by selecting a circle, segment, or distance to define its radius, and then choosing a point for its center
• demonstrate how to use a straightedge by marking a point on the circle and connecting it to the center to make a radius
• note that segment \(PQ\) is the part of the line through \(P\) and \(Q\) that has the endpoints \(P\) and \(Q\)
• note that length \(PQ\) is the distance from point \(P\) to point \(Q\)

Invite students to use their digital straightedge and compass tools to complete the remaining questions.

Student Facing

Copy this figure using only the Pen tool and no other tools. 

Familiarize yourself with your digital straightedge and compass tools by drawing a few circles of different sizes, drawing a few line segments of different lengths, and extending some of those line segments in both directions.

Copy the figure by completing these steps with the Line, Segment, and Ray tools and the Circle and Compass tools: 

• Draw a point and label it \(A\) .
• Draw a circle centered at point \(A\) with a radius equal to length \(PQ\) .
• Mark a point on the circle and label it \(B\) .
• Draw another circle centered at point \(B\) that goes through point \(A\) .
• Draw a line segment between points \(A\) and \(B\) .

Student Response

For access, consult one of our IM Certified Partners .

Give students 2 minutes of quiet work time.

• demonstrate how to use a compass by marking a point and creating a circle centered at that point

Invite students to use their tools to complete the remaining questions.

• Familiarize yourself with your straightedge and compass by drawing a few circles of different sizes, a few line segments of different lengths, and extending some of those line segments in both directions.
• Draw a circle centered at point \(A\) with a radius of length \(PQ\) .
• Draw another circle centered at point \(B\) that goes through point \(A\) .

Anticipated Misconceptions

If using rulers as straightedges, some students may wish to use the ruler to measure the length of  \(PQ\) . Emphasize that our straightedge can only be used to create lines or line segments between two marked points, but that your compass can be set to the length between two points and then moved to create a circle with that radius at any marked point.

Activity Synthesis

The goal is to make sure students understand the straightedge and compass moves that will be allowed during activities that involve constructions and why it is important to agree on standard construction moves. Ask students, “What is the difference between your attempt in the first question and what you came up with using the straightedge and compass?” (Sample response: Without the tools, it was difficult to make circles and straight lines. The compass makes it easier to make circles, and the straightedge makes it easier to make straight lines.)

Make one class display that incorporates all valid moves. This display should be posted in the classroom for the remaining lessons within this unit. It should include: 

• If starting from a blank space, start by marking two points.
• You can create a line or line segment between two marked points.
• You can create a circle centered at a marked point going through another marked point.
• You can set your compass to the length between two marked points and create a circle with that radius centered at any marked point.
• You can mark intersection points.
• You can mark a point on a circle.
• You can mark a point on a line or line segment.

Tell students that using these moves guarantees a precise construction. Conversely, eyeballing where a point or segment should go means that there is no guarantee someone will be able to reproduce it accurately.

1.2: Illegal Construction Moves (15 minutes)

Instructional Routines

• Construct It

The purpose of this activity is for students to explore why straightedge and compass constructions can be used to communicate geometric information precisely and consistently.

Identify a student who places point \(C\) closer to point \(A\) , and another student who places point \(C\) closer to point \(B\) to compare during discussion.

Arrange students in groups of 2.

• Create a circle centered at \(A\) with radius \(AB\) .
• Estimate the midpoint of segment \(AB\) , mark it with the Point on Object tool, and label it \(C\) .
• Create a circle centered at \(B\) with radius \(BC\) . Mark the 2 intersection points with the Intersection tool. Label the one toward the top of the page as \(D\) and the one toward the bottom as \(E\) .
• Use the Polygon tool to connect points \(A\) , \(D\) , and \(E\) to make triangle \(ADE\) .

For students using the digital Constructions tool, recommend that students begin by drawing a segment \(AB\) .

• Estimate the midpoint of segment \(AB\) and label it \(C\) .
• Create a circle centered at \(B\) with radius \(BC\) . This creates 2 intersection points. Label the one toward the top of the page as \(D\) and the one toward the bottom as \(E\) .
• Use your straightedge to connect points \(A\) , \(D\) , and \(E\) to make triangle \(ADE\) and lightly shade it in with your pencil.

If students do not remember how to find a midpoint, break the word down and explain that it is a point in the middle of the segment. 

The key point for discussion is that with constructions, it is possible to investigate geometry without numbers. Instead, students can use construction tools to transfer distances without measuring.

Ask students to trace triangle \(ADE\) onto tracing paper and compare their triangle with their partners. Here are some questions for discussion:

• “Which steps in the instructions made it possible for these triangles to look so different?” (Estimating the location of the midpoint.)
• “What is identical in every diagram?” (The first circle.)
• “Writing \(AD=AE\) means the length of segment \(AD\) is equal to the length of segment \(AE\) . Is that true?” (Yes, they are both radii of the same circle.)
• “Writing \(AB=2AC\) means the length of segment \(AB\) is equal to twice the length of segment  \(AC\) . Is that true?” (It looks like they might be, but we estimated the midpoint, so not necessarily.)
• “Why do valid straightedge and compass moves guarantee everyone will produce the same construction?” (There is never any estimating or eyeballing required. You are only ever using your tools to do one specific move.)

If question 2 were replaced with a method of finding the midpoint precisely with a straightedge and compass, then triangle \(ADE\) would be guaranteed to be consistent regardless of which student constructed it, up to the small error allowed by the tools. To be sure that a construction is valid, it must not include any estimation or eyeballing.

1.3: Can You Make a Perfect Copy? (10 minutes)

• MLR1: Stronger and Clearer Each Time

The purpose of this activity is to let students determine how to use straightedge and compass moves to construct a regular hexagon precisely. Students should play with construction moves until they reach their goal rather than follow an explicit demonstration of construction steps. While the term regular appears in the task, it is not important for students to know the precise definition of regular polygon at this time.

Identify students whose explanations that the sides are congruent use tracing paper, or compare the radii of the different circles in the construction. Tracing paper connects to the idea of rigid motions, while comparing radii references the precise definition of a circle, which students will use throughout this unit and subsequent units.

Arrange students in groups of 2. Give students 10 minutes of work time followed by a whole-class discussion.

Here is a hexagon with all congruent angles and all congruent sides (called a regular hexagon). 

Try to draw a copy of the regular hexagon using only the pen tool. Draw your copy next to the hexagon given, and then drag the given one onto yours. What happened?

Here is a figure that shows the first few steps to constructing the regular hexagon. Use straightedge and compass moves to finish constructing the regular hexagon. Drag the given one onto yours and confirm that it fits perfectly onto itself.

• How do you know each of the sides of the shape are the same length? Show or explain your reasoning.

Are you ready for more?

Why does the construction end up where it started? That is, how do we know the central angles go exactly 360 degrees around?

Arrange students in groups of 2. Provide access to tracing paper. Give students 10 minutes of work time followed by a whole-class discussion.

Here is a hexagon with all congruent angles and all congruent sides (called a regular hexagon).

• Draw a copy of the regular hexagon using only your pencil and no other tools. Trace your copy onto tracing paper. Try to fold it in half. What happened?

If students spend more than a few minutes without significant progress, tell them the segment given in the figure is one of the 6 sides of the hexagon. Invite students to compare the given hexagon to the start of the construction. Then ask if they can draw another segment to make an adjacent side of the hexagon.

The purpose of this discussion is to build toward the concept of a proof by asking students to informally explain why a fact about a geometric object must be true. Ask previously identified students to share their responses to “How do you know each of the sides of the shape are the same length?”

Lesson Synthesis

Point out the display of straightedge and compass moves again. Ask students to identify and define the geometric terms in the display.

• If starting from a blank space, start by marking 2  points .
• Create a line or line segment between 2 marked points.
• Create a circle centered at a marked point going through another marked point.
• Set your compass to the length between 2 marked points and create a circle with that radius centered at any marked point.
• Mark intersection points.
• Mark a point on a circle.
• Mark a point on a line or line segment.

After several students share, tell the class that point, line, and distance (or length) are undefined terms. We can use these undefined terms to define other terms. It is important to know that:

• points are infinitesimally small 
• lines are infinitely long, extending in both directions
• part of a line with one endpoint is called a ray, and it extends in one direction
• part of a line with two endpoints is called a segment, and it has a measurable length
• a circle is made up of all the points a set distance from a point
• the point is called the center, and the set distance is called the radius

Tell students that, in this course, they will build on their previous understanding of these terms and others to use precise definitions to describe geometric figures.

1.4: Cool-down - Build It (5 minutes)

Student lesson summary.

To construct geometric figures, we use a straightedge and a compass. These tools allow us to create precise drawings that someone else could copy exactly.

• We use the straightedge to draw a line segment , which is a set of points on a line with 2 endpoints. 
• We name a segment by its endpoints. Here is segment \(AB\) , with endpoints \(A\) and \(B\) .
• We use the compass to draw a circle , which is the set of all points the same distance from the center. 

We describe a circle by naming its center and radius. Here is the circle centered at \(F\) with radius \(FG\) .

Early mathematicians noticed that certain properties of shapes were true regardless of how large or small they were. Constructions were used as a way to investigate what has to be true in geometry without referring to numbers or direct measurements.

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Geometry (all content)

Unit 1: lines, unit 2: angles, unit 3: shapes, unit 4: triangles, unit 5: quadrilaterals, unit 6: coordinate plane, unit 7: area and perimeter, unit 8: volume and surface area, unit 9: pythagorean theorem, unit 10: transformations, unit 11: congruence, unit 12: similarity, unit 13: trigonometry, unit 14: circles, unit 15: analytic geometry, unit 16: geometric constructions, unit 17: miscellaneous.

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1.1E: Exercises for Section 1.1

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• Page ID 69574

• Gilbert Strang & Edwin “Jed” Herman

For exercises 1 - 6, (a) determine the domain and the range of each relation, and (b) state whether the relation is a function.

a. Domain = {\(−3,−2,−1,0,1,2,3\)}, Range = {\(0,1,4,9\)}

b. Yes, a function

a. Domain = {\(0,1,2,3\)}, Range = {\(−3,−2,−1,0,1,2,3\)}

b. No, not a function

a. Domain = {\(3,5,8,10,15,21,33\)}, Range = {\(0,1,2,3\)}

For exercises 7 - 13, find the values for each function, if they exist, then simplify.

a. \(f(0)\) b. \(f(1)\) c. \(f(3)\) d. \(f(−x)\) e. \(f(a)\) f. \(f(a+h)\)

7) \(f(x)=5x−2\)

8) \(f(x)=4x^2−3x+1\)

9) \(f(x)=\dfrac{2}{x}\)

10) \(f(x)=|x−7|+8\)

11) \(f(x)=\sqrt{6x+5}\)

12) \(f(x)=\dfrac{x−2}{3x+7}\)

13) \(f(x)=9\)

For exercises 14 - 21, find the domain, range, and all zeros/intercepts, if any, of the functions.

14) \(f(x)=\dfrac{x}{x^2−16}\)

15) \(g(x)=\sqrt{8x−1}\)

16) \(h(x)=\dfrac{3}{x^2+4}\)

17) \(f(x)=−1+\sqrt{x+2}\)

18) \(f(x)=1x−\sqrt{9}\)

19) \(g(x)=\dfrac{3}{x−4}\)

20) \(f(x)=4|x+5|\)

21) \(g(x)=\sqrt{\dfrac{7}{x−5}}\)

For exercises 22 - 27, set up a table to sketch the graph of each function using the following values: \(x=−3,−2,−1,0,1,2,3.\)

22) \(f(x)=x^2+1\)

23) \(f(x)=3x−6\)

24) \(f(x)=\frac{1}{2}x+1\)

25) \(f(x)=2|x|\)

26) \(f(x)=-x^2\)

27) \(f(x)=x^3\)

For exercises 28 - 35, use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

a. Domain and range

b. \(x\) -intercept, if any (estimate where necessary)

c. \(y\)-Intercept, if any (estimate where necessary)

d. The intervals for which the function is increasing

e. The intervals for which the function is decreasing

f. The intervals for which the function is constant

g. Symmetry about any axis and/or the origin

h. Whether the function is even, odd, or neither

For exercises 36 - 41, for each pair of functions, find a. \(f+g\) b. \(f−g\) c. \(f⋅g\) d. \(f/g\). Determine the domain of each of these new functions.

36) \(f(x)=3x+4,\quad g(x)=x−2\)

37) \(f(x)=x−8,\quad g(x)=5x^2\)

38) \(f(x)=3x^2+4x+1,\quad g(x)=x+1\)

39) \(f(x)=9−x^2,\quad g(x)=x^2−2x−3\)

40) \(f(x)=\sqrt{x},\quad g(x)=x−2\)

41) \(f(x)=6+\dfrac{1}{x},\quad g(x)=\dfrac{1}{x}\)

a. \(6+\dfrac{2}{x};\quad x≠0\)

b. \(6; \quad x≠0\)

c. \(6x+\dfrac{1}{x^2};\quad x≠0\)

d. \(6x+1;\quad x≠0\)

For exercises 42 - 48, for each pair of functions, find a. \((f∘g)(x)\) and b. \((g∘f)(x)\) Simplify the results. Find the domain of each of the results.

42) \(f(x)=3x,\quad g(x)=x+5\)

43) \(f(x)=x+4,\quad g(x)=4x−1\)

44) \(f(x)=2x+4,\quad g(x)=x^2−2\)

45) \(f(x)=x^2+7,\quad g(x)=x^2−3\)

46) \(f(x)=\sqrt{x}, \quad g(x)=x+9\)

47) \(f(x)=\dfrac{3}{2x+1},\quad g(x)=\dfrac{2}{x}\)

a. \(\dfrac{3x}{4+x};\quad x≠0,−4\)

b. \(\dfrac{4x+2}{3};\quad x≠−\frac{1}{2}\)

48) \(f(x)=|x+1|,\quad g(x)=x^2+x−4\)

49) The table below lists the NBA championship winners for the years 2001 to 2012.

a. Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.

b. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.

50) [T] The area \(A\) of a square depends on the length of the side s.

a. Write a function \(A(s)\) for the area of a square.

b. Find and interpret \(A(6.5)\).

c. Find the exact and the two-significant-digit approximation to the length of the sides of a square with area 56 square units.

51) [T] The volume of a cube depends on the length of the sides \(s.\)

a. Write a function \(V(s)\) for the area of a square.

b. Find and interpret \(V(11.8)\).

52) [T] A rental car company rents cars for a flat fee of $20 and an hourly charge of $10.25. Therefore, the total cost \(C\) to rent a car is a function of the hours \(t\) the car is rented plus the flat fee.

a. Write the formula for the function that models this situation.

b. Find the total cost to rent a car for 2 days and 7 hours.

c. Determine how long the car was rented if the bill is $432.73.

53) [T] A vehicle has a 20-gal tank and gets 15 mpg. The number of miles \(N\) that can be driven depends on the amount of gas \(x\) in the tank.

a. Write a formula that models this situation.

b. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) 3/4 of a tank of gas.

c. Determine the domain and range of the function.

d. Determine how many times the driver had to stop for gas if she has driven a total of 578 mi.

54) [T] The volume \(V\) of a sphere depends on the length of its radius as \(V=(4/3)πr^3\). Because Earth is not a perfect sphere, we can use the mean radius when measuring from the center to its surface. The mean radius is the average distance from the physical center to the surface, based on a large number of samples. Find the volume of Earth with mean radius \(6.371×106\) m.

55) [T] A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by \(r(t)=6−\dfrac{5}{t^2+1}\), where \(t\) is time measured in hours since a circle of a 1-cm radius of the bacterium was put into the culture.

a. Express the area of the bacteria as a function of time.

b. Find the exact and approximate area of the bacterial culture in 3 hours.

c. Express the circumference of the bacteria as a function of time.

d. Find the exact and approximate circumference of the bacteria in 3 hours.

56) [T] An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79x\), where \(x\) is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245x\), where \(x\) is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars.

a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process?

b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra $200 when she arrived in Paris.

57) [T] The manager at a skateboard shop pays his workers a monthly salary \(S\) of $750 plus a commission of $8.50 for each skateboard they sell.

a. Write a function \(y=S(x)\) that models a worker’s monthly salary based on the number of skateboards \(x\) he or she sells.

b. Find the approximate monthly salary when a worker sells 25, 40, or 55 skateboards.

c. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of $1400. (Hint: Find the intersection of the function and the line \(y=1400\).)

58) [T] Use a graphing calculator to graph the half-circle \(y=\sqrt{25−(x−4)^2}\). Then, use the INTERCEPT feature to find the value of both the \(x\)- and \(y\)-intercepts.

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org .

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Home > CCG > Chapter 6 > Lesson 6.1.1

Lesson 6.1.1, lesson 6.1.2, lesson 6.1.3, lesson 6.1.4, lesson 6.1.5, lesson 6.2.1, lesson 6.2.2, lesson 6.2.3, lesson 6.2.4, lesson 6.2.5, lesson 6.closure.

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