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K12MATH010: Geometry

Unit 4: Connecting Algebra and Geometry through Coordinates   In this unit, you will further examine the properties you have learned thus far but with the help of algebra and the coordinate plane. You will use the Pythagorean theorem to help you find distances and your knowledge of equations of lines to verify geometric relationships, including properties of triangles and quadrilaterals, slopes of parallel and perpendicular lines, and special lines within triangles. Are you feeling a bit unsure of your algebra skills? No fear, as there will be brief reviews of the skills needed in order to apply algebra to the investigation of geometric properties![[4]](#_ftn1)


[4]Ibid.

Unit 4 Time Advisory
Complething this unit should take you approximately 15 hours.

☐    Subunit 4.1: 2 hours, 30 minutes

☐    Subunit 4.2: 8 hours, 30 minutes

☐    Subunit 4.3: 4 hours

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:
- Use coordinates to prove simple geometric theorems algebraically. - Use coordinates to find the area and perimeter of figures using the distance formula. - Determine whether lines are parallel or perpendicular using the coordinates of points and algebra. - Identify and graph special lines of a triangle and determine their point of concurrence.

 
Standards Addressed (Common Core): - CCSS.Math.Content.HSG-GPE.B.4 - CCSS.Math.Content.HSG-GPE.B.5 - CCSS.Math.Content.HSG-GPE.B.6 - CCSS.Math.Content.HSG-GPE.B.7 - CCSS.ELA-Literacy.RST.9-10.3 - CCSS.ELA-Literacy.RST.9-10.5 - CCSS.ELA-Literacy.WHST.9-10.1d - CCSS.ELA-Literacy.WHST.9-10.1e

4.1 Parallel and Perpendicular Lines in the Coordinate Plane   This subunit is the first that examines geometric properties using algebra and coordinate geometry. Here you will review how to calculate slope and write the equation of a line. With this information, you will be well armed to investigate the slopes of parallel and perpendicular lines, which will allow you to classify quadrilaterals on the coordinate plane in later subunits. Here you will be able to revisit proofs and apply logic skills learned in prior units to your understanding of the properties and relationships of quadrilaterals. The resources in this section provide explanatory content and practice problems.

4.1.1 Review: Slope   - Explanation: CK-12: Geometry: “Parallel and Perpendicular Lines in the Coordinate Plane” Link: CK-12: Geometry: “Parallel and Perpendicular Lines in the Coordinate Plane” (HTML)
 
Instructions: Please click on the link above, and read the section entitled “Slope in the Coordinate Plane.” Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following question: How is slope calculated?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●  
  [CCSS.Math.Content.HSG-GPE.B.4](http://www.corestandards.org/Math/Content/HSG/GPE/B/4)  
 ●  
  [CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

4.1.2 Review: Equations of Lines   - Explanation: CK-12 Algebra: “Linear Equations in Slope-Intercept Form”,“Linear Equations in Point-Slope Form”and “Linear Equations in Standard Form” Link: CK-12 Algebra: “Linear Equations in Slope-Intercept Form”, “Linear Equations in Point-Slope Form” and “Linear Equations in Standard Form” (HTML)
 
Instructions: Please click on each link, and read the sections listed above. Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What are the different forms for writing the equation of a line? How do you change the equation of line from one form to another? How do you identify the slope of a line in each form of the equation of a line?
 
Reading these sections, taking notes, and completing the example problems should take approximately 60 minutes.
 
Standards Addressed (Common Core):

 ●    
 [CCSS.Math.Content.HSG-GPE.B.4](http://www.corestandards.org/Math/Content/HSG/GPE/B/4)  
 ●    
 [CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

4.1.3 Slopes of Parallel and Perpendicular Lines   - Explanation: CK-12: Geometry: “Parallel and Perpendicular Lines in the Coordinate Plane” Link: CK-12: Geometry: “Parallel and Perpendicular Lines in the Coordinate Plane” (HTML)
 
Instructions: Please click on the link above, and read the sections entitled “Slopes of Parallel Lines,” “Slopes of Perpendicular Lines,” and “Graphing Parallel and Perpendicular Lines.” Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the relationship among the slopes of parallel lines? What is the relationship among the slopes of perpendicular lines? How do you identify the slope of a line, given the equation of the line? How do you graph these lines?
 
Reading these sections, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.4](http://www.corestandards.org/Math/Content/HSG/GPE/B/4)  
 ●     
[CCSS.Math.Content.HSG-GPE.B.5](http://www.corestandards.org/Math/Content/HSG/GPE/B/5)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Equations of Parallel and Perpendicular Lines” Link: Khan Academy’s “Equations of Parallel and Perpendicular Lines” (HTML)
     
    Instructions: Please click on the link above, which provides a series of practice problems that allow you to determine if lines are parallel or perpendicular and to graph parallel and perpendicular lines. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.4
    ●      CCSS.Math.Content.HSG-GPE.B.5
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Checkpoint: Mathematics Assessment Project: “Finding Equations of Parallel and Perpendicular Lines” Link: Mathematics Assessment Project: “Finding Equations of Parallel and Perpendicular Lines” (HTML)
     
    Instructions: Please click on the link above, and scroll down to the section of the lesson titled “Resources.” Download the PDF file underneath “Lesson (complete).” The assessment is on page S-1 and is titled “Parallel and Perpendicular Lines.” Complete the questions to the best of your ability. When you are finished, use the solutions on page T-7 to check your understanding.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●    CCSS.Math.Content.HSG-GPE.B.4

    ●    CCSS.Math.Content.HSG-GPE.B.5
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2 Area and Perimeter   This subunit continues with the task of analyzing geometric properties using algebra and coordinate geometry. Here you will review how to calculate the distance of a segment, find the midpoint of a segment, and solve systems of equations. With this information, you will be able to identify the key dimensions of figures on the coordinate plane, calculate their distances, and find the area and perimeter of these figures. These skills are helping you build up to finding the areas and perimeters of irregular (curved) figures, as seen in calculus and used by engineers when developing many products, including cars. The resources in this subunit provide explanatory content, practice problems, and investigatory work.

4.2.1 Algebra Review: Distance Formula   - Explanation: CK-12: Geometry: “The Distance Formula” Link: CK-12: Geometry: “The Distance Formula” (HTML)
 
Instructions: Please click on the link above, and read the section. Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following question: What is the process for calculating the length of a line on the coordinate plane?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Distance Formula” Link: Khan Academy’s “Distance Formula” (HTML)
     
    Instructions: Please click on the link above, which provides a series of problems that allow you to practice calculating the distance of line segments. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.2 Algebra Review: Midpoint Formula   - Explanation: CK-12: Geometry: “Midpoints and Bisectors” Link: CK-12: Geometry: “Midpoints and Bisectors” (HTML)
 
Instructions: Please click on the link above, and read the section entitled “Midpoint Formula.” Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following question: What is the process for identifying the midpoint of a line segment on the coordinate plane?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.6](http://www.corestandards.org/Math/Content/HSG/GPE/B/6)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Midpoint Formula” Link: Khan Academy’s “Midpoint Formula” (HTML)
     
    Instructions: Please click on the link above, which provides a series of problems that allow you to practice calculating the midpoint of line segments. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.6
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.3 Algebra Review: Intersecting Lines   - Explanation: CK-12 Algebra: “Linear Systems by Graphing,” “Solving Linear Systems by Substitution,” and “Solving Linear Systems by Elimination through Addition and Subtraction” Link: CK-12 Algebra: “Linear Systems by Graphing,” “Solving Linear Systems by Substitution,” and “Solving Linear Systems by Elimination through Addition and Subtraction” (HTML)
 
Instructions: Please click on the links, and read each section. Take notes and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What are the three ways to determine the point of concurrency (the point of intersection) of two lines? Which process do you find the easiest? Can you always solve a linear system by graphing? Why or why not?
 
Reading these sections, taking notes, and completing the example problems should take approximately 1 hour.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSA-REI.B.3](http://www.corestandards.org/Math/Content/HSA/REI/B/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: Khan Academy’s “Systems of Equations” Link: Khan Academy’s “Systems of Equations” (HTML)
     
    Instructions: Please click on the link above, which provides a series of problems that allow you to practice solving systems of equations using a variety of methods. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.6
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.4 Parallelograms   - Explanation: CK-12: Geometry: “Proving Quadrilaterals are Parallelograms” Link: CK-12: Geometry: “Proving Quadrilaterals are Parallelograms” (HTML)
 
Instructions: Please click on the link above, and scroll down to the section entitled “Showing a Quadrilateral is a Parallelogram in the Coordinate Plane.” Read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the process for determining if a quadrilateral in the coordinate plane is a parallelogram? What information do you need to know? How do you find this information on the coordinate plane?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.

 Standards Addressed (*Common Core*):  

 ●     
[CCSS.Math.Content.HSG-GPE.B.4](http://www.corestandards.org/Math/Content/HSG/GPE/B/4)  
 ●     
[CCSS.Math.Content.HSG-GPE.B.5](http://www.corestandards.org/Math/Content/HSG/GPE/B/5)  
 ●     
[CCSS.Math.Content.HSG-GPE.B.6](http://www.corestandards.org/Math/Content/HSG/GPE/B/6)  
 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Web Media: SOPHIA: Mr. Hennen’s “Coordinate Geometry of Quadrilaterals” Link: SOPHIA: Mr. Hennen’s “Coordinate Geometry of Quadrilaterals” (HTML)
     
    Instructions: Please click on the link above, read the content, and watch the videos that explain how to prove which type of parallelogram is on the coordinate plane. Take notes on each video so that you can apply this knowledge in the next activity.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: GeoGebra: “Quadrilaterals Lab” Link: GeoGebra: “Quadrilaterals Lab” (Java)
     
    Instructions: Please click on the link above. Use the content you learned in the readings and videos from the previous subunit to investigate the quadrilaterals and determine if they are special parallelograms, and if so, which ones. Challenge yourself to write a two-column proof that demonstrates why a figure is the special parallelogram that you identify.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.5
    ●      CCSS.ELA-Literacy.WHST.9-10.1d
    ●      CCSS.ELA-Literacy.WHST.9-10.1e
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.5 Perimeter   - Explanation: CK-12: Geometry: “Triangles and Parallelograms” Link: CK-12: Geometry: “Triangles and Parallelograms” (HTML)
 
Instructions: Please click on the link above, and scroll down to the section entitled “Areas and Perimeters of Squares and Rectangles.” Read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the process for calculating the perimeter of a figure? How would this work on the coordinate plane? If you had the coordinates of the vertices of a figure, how would you calculate the perimeter? What information would you need?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Did I Get This? Activity: GeoGebra: “Triangle Perimeter Calculations” Link: GeoGebra: “Triangle Perimeter Calculations” (Java)
     
    Instructions: Please click on the link above. Here you will find the link to an applet that allows you to sketch different triangles on the coordinate plane. Click on “Go to Student Worksheet” to begin. Sketch a triangle, using the applet. With the distance formula, calculate the length of each side of the triangle and then the perimeter of the entire triangle. Compare your answer to the computer-generated answer. Repeat this activity three times.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.6 Area of a Triangle   - Explanation: CK-12: Geometry: “Triangles and Parallelograms” Link: CK-12: Geometry: “Triangles and Parallelograms” (HTML)
 
Instructions: Please click on the link above, and scroll down to the section entitled “Area of a Triangle.” Read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the process for calculating the area of a triangle? How would this work on the coordinate plane? If you had the coordinates of the vertices, how would you calculate the area? What information would you need?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Web Media: Math Open Reference: “Area of a Triangle by Formula (Coordinate Geometry)” Link: Math Open Reference: “Area of a Triangle by Formula (Coordinate Geometry)” (Flash)
     
    Instructions: Please click on the link above, read the content, and take notes. Then scroll down to where it says “Things to Try.” Follow the instructions that will allow you to practice calculating the area of a triangle on the coordinate plane.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.5
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: Math Open Reference: “Area of a Triangle—Box Method” Link: Math Open Reference: “Area of a Triangle—Box Method” (Flash)
     
    Instructions: Please click on the link above, read the content, and take notes. Then scroll down to where it says “Things to Try.” Follow the instructions that will allow you to practice calculating the area of a triangle on the coordinate plane using another method.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.5
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: Math Open Reference: “Heron’s Formula for the Area of a Triangle” Link: Math Open Reference: “Heron’s Formula for the Area of a Triangle” (Flash)
     
    Instructions: Please click on the link above, read the brief content, and investigate Heron’s formula following the instructions of “Try This.” Then scroll down to where it says “Test Your Knowledge.” Here you will find two examples to try. The answers are there for you to revise your work. Additionally, upon completion of this activity, take time to summarize the different ways in which you have learned to calculate the area of a triangle on the coordinate plane. Is there one method you prefer over others? Why? Does the preferred method change depending on the triangle? Explain.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.5
    ●      CCSS.ELA-Literacy.WHST.9-10.1d
    ●      CCSS.ELA-Literacy.WHST.9-10.1e
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Did I Get This? Activity: Khan Academy’s “Heron’s Formula” Link: Khan Academy’s “Heron’s Formula” (HTML)
     
    Instructions: Please click on the link above, which provides a series of problems that allow you to practice calculating the area of a triangle using Heron’s formula. Each question has explanations for the example, worked out step-by-step if you need hints along the way. The program monitors your success, and when you have correctly completed six questions consecutively, it will tell you that you are ready to move on to a new skill.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.2.7 Area of a Parallelogram   - Explanation: CK-12: Geometry: “Triangles and Parallelograms” Link: CK-12: Geometry: “Triangles and Parallelograms” (HTML)
 
Instructions: Please click on the link above, and scroll down to the section entitled “Area of a Parallelogram.” Read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is the process for calculating the area of a parallelogram? How would this work on the coordinate plane? If you had the coordinates of the vertices, how would you calculate the area? What information would you need?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Web Media: Math Open Reference: “Parallelogram (Coordinate Geometry)” Link: Math Open Reference: “Parallelogram (Coordinate Geometry)” (Flash)
     
    Instructions: Please click on the link above, and read the content explaining how to calculate the height and width of the parallelogram. Next, scroll down to “Things to Try.” Follow the instructions but take the activity one step further by using your calculations of the width and the height to calculate the area.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
    ●      CCSS.ELA-Literacy.RST.9-10.3
    ●      CCSS.ELA-Literacy.RST.9-10.5
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Checkpoint: Mathematics Vision Project: “Secondary One Mathematics: An Integrated Approach, Module 6 (Honors), Connecting Algebra and Geometry” Link: Mathematics Vision Project: “Secondary One Mathematics: An Integrated Approach, Module 6 (Honors), Connecting Algebra and Geometry” (PDF)
     
    Instructions: Please click on the link above, and scroll down to page 24. On this page you will find six figures on the coordinate plane. The activity asks you to find the perimeter of each figure. Challenge yourself to find both the perimeter and the area for each one. At this point in the course we have not discussed circles, but you should be familiar with the formulas for circumference and for area of circles. After you have completed each figure, click here to check your answers.
     
    Completing this activity should take approximately 1 hour.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-GPE.B.7
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.3 Lines in Triangles   This subunit explores the important lines within a triangle and the points of concurrency that they mark. You will discover the important properties of these points of concurrency that have interesting implications for physics and other sciences. The resources in this section provide explanatory content, practice problems, and investigatory work.

4.3.1 Midsegments of a Triangle   - Explanation: CK-12: Geometry: “Midsegments of a Triangle” Link: CK-12: Geometry: “Midsegments of a Triangle” (HTML)
 
Instructions: Please click on the link above, read the section, take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is a midsegment of a triangle? What are some of its important characteristics? How do you find the midsegment of a triangle without coordinate geometry? How do you find the midsegment of a triangle with coordinate geometry?
 
Reading this section, taking notes, and completing the example problems should take approximately 45 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-GPE.B.4](http://www.corestandards.org/Math/Content/HSG/GPE/B/4)  
 ●     
[CCSS.Math.Content.HSG-GPE.B.7](http://www.corestandards.org/Math/Content/HSG/GPE/B/7)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Web Media: GeoGebra: “Midsegments of Triangles” Link: GeoGebra: “Midsegments of Triangles” (Java)
     
    Instructions: Please click on the link above. Use the program to explore how the slope and length of the midsegment changes in relation to the slope and length of the slides. Take this activity a step further by drawing three triangles on a coordinate plane and using coordinate geometry to graph the midsegment. Find the midpoints of each side and connect them. What is the equation of the line of the midsegment in each of the triangles that you drew?
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.3.2 Medians of a Triangle   - Explanation: CK-12: Geometry: “Medians and Altitudes in Triangles” Link: CK-12: Geometry: “Medians and Altitudes in Triangles” (HTML)
 
Instructions: Please click on the link above, and scroll down to read the sections titled “Medians” and “Point of Concurrency for Medians.” Take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is a median of a triangle? What are some of its important characteristics? How do you find the median of a triangle without coordinate geometry? How do you find the median of a triangle with coordinate geometry?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Activity: Math Open Reference: “Triangle Centroid Definition” Link: Math Open Reference: “Triangle Centroid Definition” (Flash)
     
    Instructions: Please click on the link above, and read about the relationship between the medians of a triangle and the centroid of a triangle. Scroll down to where it says “Things to Try.” Here you will find six steps for an activity that will allow you to explore the triangle center created by the intersection of medians and understand its value. Complete the activity. This page includes information about other triangle centers as well and provides links for additional activities for students interested in exploring further.
     
    Completing this activity should take approximately 30 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

  • Web Media: GeoGebra: “Basic Investigation of Medians in a Triangle” Link: GeoGebra: “Basic Investigation of Medians in a Triangle” (Java)
     
    Instructions: Please click on the link above, follow the instructions of the activity, and answer the questions below the applet. Now draw a triangle on a coordinate plane and use coordinate geometry to draw the three medians of a triangle and find the centroid. Do this two or three times or until you feel comfortable with this skill. Remember, you will need to find the equation of the line for two of the medians and find their point of intersection by solving a system of equations.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.3.3 Altitudes of a Triangle   - Explanation: CK-12: Geometry: “Medians and Altitudes in Triangles” Link: CK-12: Geometry: “Medians and Altitudes in Triangles” (HTML)
 
Instructions: Please click on the link above, and scroll down and read the section titled “Altitudes.” Take notes, and complete the example problems as you go. The solutions are explained following each problem. As you read, consider the following questions: What is an altitude of a triangle? What are some of its important characteristics? How do you find the altitude of a triangle without coordinate geometry? How do you find the altitude of a triangle with coordinate geometry?
 
Reading this section, taking notes, and completing the example problems should take approximately 30 minutes.
 
Standards Addressed (Common Core):

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1d](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/d)  
 ●     
[CCSS.ELA-Literacy.WHST.9-10.1e](http://www.corestandards.org/ELA-Literacy/WHST/9-10/1/e)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
  • Web Media: GeoGebra: “Orthocenter of a Triangle” Link: GeoGebra: “Orthocenter of a Triangle” (Java)
     
    Instructions: Please click on the link above, follow the instructions of the activity, and answer the questions below the applet. Now draw a triangle on a coordinate plane and use coordinate geometry to draw the three altitudes of the triangle and find the orthocenter. Do this two or three times or until you feel comfortable with this skill. Remember, you will need to find the equation of the line for two of the altitudes and find their point of intersection by solving a system of equations.
     
    Completing this activity should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

    ●      CCSS.Math.Content.HSG-CO.C.9
    ●      CCSS.Math.Content.HSG-CO.C.10
     
    Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

4.3.4 Points of Concurrence   - Web Media: Math Open Reference: “Euler Line” Link: Math Open Reference: “Euler Line” (Flash)

 Instructions: Please click on the link above, and watch the short
demonstration of the points of concurrency—centroid, circumcenter,
and orthocenter align to create the Euler line. After investigating
the Euler Line, draw a triangle on the coordinate plane. Then, using
coordinate geometry, find the centroid, circumcenter, and
orthocenter, and draw a line through these three points, creating a
Euler line.  
    
 Completing this activity should take approximately 30 minutes.  
    
 Standards Addressed (*Common Core*):  

 ●     
[CCSS.Math.Content.HSG-CO.C.9](http://www.corestandards.org/Math/Content/HSG/CO/C/9)  
 ●     
[CCSS.Math.Content.HSG-CO.C.10](http://www.corestandards.org/Math/Content/HSG/CO/C/10)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)  
 ●     
[CCSS.ELA-Literacy.RST.9-10.5](http://www.corestandards.org/ELA-Literacy/RST/9-10/5)  
    
 Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.