Fall 2008 MATH 330A Foundations of Geometry I

Assigned Homework Set #1: Drawings for Textbook Problem Set 0

In this first homework set, you will produce drawings to accompany Problem Set 0 on pages 6-7 of the textbook.

Important information about the assignment and about this web page.

• None of the problems in Problem Set 0 specify lengths or angles (except for problem #6, which specifies only that the triangle have a right angle.). Therefore, certain parts of each drawing must be moveable in order to illustrate any configuration of the general type described in the problem statement.
• You will use a computer program called GeoGebra. It has functionality similar to the popular Geometer's SketchPad program, but GeoGebra is free and can be run on your home computer. It has the added advantage that it can export drawings in a variety of forms, including HTML.
• Your homework submissions in this course will include a mixture of paper and electronic forms. In this first homework set you will submit everything electronically: you will create drawings in GeoGebra, save them in a folder, zip the folder, and finally e-mail me the zipped folder.

• Login to the computer
  • id: your Oak ID
  • password: last 6 digits of your student ID number
• Filing task
  • On your computer desktop, create a folder called H1.Lastname, using your last name.

Problem [1] (5 points) The goal is to create a drawing for the Treasure Island Problem discussed in the text and in problem 1 of Problem Set 0.

• Open a new GeoGebra document.
• Create points to represent the Coconut Tree, the Banana Tree, and the Gallows. Label the three points C, B, and G
• Create a point to represent the spike obtained by walking from the Gallows to the Coconut Tree, turning right 90 degrees, and walking the same distance again. (Hint: use the tool called Rotate object around point by angle.) Label this point S1.
• Create a point to represent the spike obtained by walking from the Gallows to the Banana Tree, turning left 90 degrees, and walking the same distance again. Label this point S2.
• Create a point to represent the treasure. (This point will be halfway between the spikes.) Label this point T.
• Create line segments GC, CS1, GB, BS2, and S1S2. Color the two segments GC and CS1 one color. Color the two segments GB and BS2 another color. Color segment S1S2 a third color.
• In your drawing, points C, B, and G should be free and all other parts of the drawing should be dependent. Move these three points around and observe how the treasure moves.
• In your own words, conjecture the solution to the Treasure Island Problem. That is, conjecture how you can locate the treasure using only the Coconut Tree and the Banana Tree. Somewhere in the drawing, insert a textbox containing text that explains your conjecture.
• Add lines and measurements to your drawing to support your conjecture.
• Somewhere in the upper right of the drawing, insert a textbox with the text H1.1.Lastname , using your last name.
• Save the drawing. Give it the filename H1.1.Lastname and put it in the folder called H1.Lastname that you created on the desktop.

Problem 2 in Problem Set 0 of the textbook is about finding the center of a circle or a circular arc drawn on a transparent piece of paper. On your own, think about an answer to that problem and compare your answer to the answer in the back of the book.

Realize that the exercise in the book has to do with technology of a sort (transparent paper), and the problem and its solution are tied to that technology. In this homework set, we will deal with another problem related to circles. The solution to the problem will depend on a different technology.

It is a fact in Euclidean Geometry that given any three non-collinear points, there exists exactly one circle that contains all three points. Here is the problem: Given three non-collinear points, draw the circle that contains all three points.

In GeoGebra, there is a no-brainer solution to this problem. But in other drawing programs, such as Geometer's SketchPad, the solution is harder. We will explore both kinds of solutions in GeoGebra.

• Open a new GeoGebra document.

• Create three non-collinear points. Label the three points A, B, and C.
• Using the Circle through three points. command, construct a circle that contains all three points.
• In your drawing, points A, B, and C should be free and the circle should be dependent.

• Create three non-collinear points somewhere outside of your circle.. Label the three points D, E, and F.
• Create line segments DE and EF.
• Create perpendicular bisectors of line segments DE and EF.
• The two perpendicular bisectors will intersect. Label their point of intersection G.
• Using the Circle with Center Through Pointcommand, create a circle centered at G and passing through point D. Observe that the circle also contains points E and F.
• In your drawing, points D, E, and F should be free and all other parts should be dependent.
• Somewhere in the upper right of the drawing, insert a textbox with the text H1.2.Lastname , using your last name.
• Save the drawing. Give it the filename H1.2.Lastname and put it in the folder called H1.Lastname that you created on the desktop.

• Open a new GeoGebra document.
• Create a drawing like Figure 0.9 on page 7 of the textbook. Be sure that the vertices of the convex quadrilateral are labeled A, B, C, and D. Label the centers of the squares C1, C2, C3, and C4. (Don't bother trying to use subscripts in your labels.)
• Conjecture the relationship between the two red line segments. Somewhere in the drawing, insert a textbox containing text that explains your conjecture in your own words.
• Add measurements to your drawing to support your conjecture.
• In your drawing, points A, B, C, and D should be free and all other parts of the drawing should be dependent.
• Somewhere in the upper right of the drawing, insert a textbox with the text H1.3.Lastname , using your last name.
• Save the drawing. Give it the filename H1.3.Lastname and put it in the folder called H1.Lastname that you created on the desktop.

• Open a new GeoGebra document.
• Create a triangle with vertices A, B, and C. Color the sides and vertices black and the interior dark gray.
• Create lines AB, BC, and CA. Make them thick black dotted lines.
• Create the three altitudes of the triangle. Make them thick red dotted lines.
• Create a point at the intersection of the three red altitudes. Color it red and give it the label H.
• Create a point at the foot of each altitude. Color them blue.
• Put the label F1 on the foot of the altitude from vertex A, the label F2 on the foot of the altitude from vertex B, and so on.
• Create midpoints of segments AB, BC, CA, AH, BH, and CH. Color them blue.
• Put the labels M1, M2, and M3 on the midpoints of segments BC, CA, and AB, respectively. (this is not the same as the book's labeling).
• Put the labels N1, N2, and N3 on the midpoints of segments AH, BH, and CH, respectively.
• Notice that you have created nine blue points in all. Create a green circle that goes through three of them. Observe that it will automatically go through all nine blue points.
• In your drawing, points A, B, and C should be free and all other parts of the drawing should be dependent.
• Somewhere in the upper right of the drawing, insert a textbox with the text H1.4.Lastname , using your last name.
• Save the drawing. Give it the filename H1.4.Lastname and put it in the folder called H1.Lastname that you created on the desktop.

• Open a new GeoGebra document.
• Create a drawing like Figure 0.3 on page 3 of the textbook. Be sure to label the points of your drawing using the same labeling scheme as Figure 0.3
• Be sure that your drawing includes measurements that show that triangle DEF is equilateral. Notice that as you move points A, B, and C around, the triangle DEF remains equilateral.
• In your drawing, points A, B, and C should be free and all other parts of the drawing should be dependent.
• Somewhere in the upper right of the drawing, insert a textbox with the text H1.5.Lastname , using your last name.
• Save the drawing. Give it the filename H1.5.Lastname and put it in the folder called H1.Lastname that you created on the desktop.

• Filing task
  • The folder called H1.Lastname that you created on the desktop should now contain your five completed GeoGebra drawings, named H1.1.Lastname through H1.5.Lastname.
  • Right-click on this folder. A pop-up menu will appear. On that menu, click on the item called 7-zip. Another little menu will pop-up to the side of the menu already open. There will only be one item on this menu. Click on the menu item called Add to archive...
  • A dialog box called Add to Archive will pop up. It will be suggesting an archive name of H1.Lastname.7z. Click OK to accept this name. A file called H1.Lastname.7z should appear on your desktop.
• e-mail task
  • Using your OU e-mail account send me an e-mail
    • Recipients:
      • me: Mark.Barsamian.1@ohio.edu
      • you: your OU e-mail address
    • Subject Line: Math 330A H1 Lastname
    • Attachment: the file H1.Lastname.7z that is on your desktop
    • Body of the message: Math 330A Homework from Your Name
  • Be sure to use your OU e-mail account for all transactions in this course, and be sure to include both you and me on the list of recipients. If an e-mail that you send to both me and you arrives successfully at your inbox, you can be reasonably assured that the e-mail also arrived at my inbox.
• Logout
  • Close all programs and logout.

Last updated September 6, 2008

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