Department of Mathematics

Winter 2009 MATH 330B/539 Geometry Homework 5 Hints

• due Monday, February 16, 2009
• 5.2 # 2(G), 5a, 10
• 5.3 # 3(G), 4(G)
• Also do Geogebra drawings for the problems with a (G).

Hints for 5.2#2 written solution

• So that we can all compare our solutions, let's agree on some labeling. Starting at the lower left vertex and proceeding counterclockwise, label the six vertices of the polygon A, C, D, E, F, G.
• Points B (the ball) and H (the hole) are given points.
• The goal is to describe the locations of points P, Q, and S such that the length of the path B-P-Q-S-H is minimized.
• Let's take the approach that the book uses in such problems. Namely, suppose that points P, Q, and S are in their optimal position. In that case, the following paths all have the same lengths:
  • B-P-Q-S-H
  • B'-P-Q-S-H
  • B'-Q-S-H
  • B''-Q-S-H
  • B''-S-H
  • B'''-S-H
  • B'''-H
• Explain what points B', B'', and B''' I am talking about. Find a reference for my claim that those paths have the same length. Explain how the points B', B'', and B''' can be used to get the locations of points P, Q, and S.

Preparation for Geogebra

• Create a folder called Geometry.H5.Lastname on your computer desktop or on your Thumbdrive.

Instructions for 5.2#2 Geogebra Drawing

• Points A, C, D, E, and F are free but are constrained to move in a way that maintains the right angles shown in the figure in the book.
• Point G is dependent.
• Points B and H are free.
• Points P, Q, and S are dependent, and are located at the optimal spots that you learned about in your written solution of the problem.
• The segmented path B-P-Q-S-H is shown
• The length of segmented path B-P-Q-S-H is displayed as a quantity in your list of objects.
• Three more points U, V, and W are free to move on segments AC, CD, and DE respectively.
• The segmented path B-U-V-W-H is shown
• The length of segmented path B-U-V-W-H is displayed as a quantity in your list of objects.

Steps in the construction

• Build Polygon(ACDEFG).
  • Create three points. They will be free, and Geogebra will label them A,B,C.
  • Create line(AC). Geogebra will give it some label that doesn't matter.
  • Create a line through point C perpendicular to line(AC).
  • On this new line, create a point D.
  • Create a line through point D perpendicular to line(CD).
  • On this new line, create a point E.
  • Create a line through point E perpendicular to line(DE).
  • On this new line, create a point F.
  • Create a line through point F perpendicular to line(EF).
  • Create a line through point A perpendicular to line(AC).
  • Create a point G at the intersection of these to most recent lines.
  • Now that you have points A through G, create a polygon(ACDEFG).
  • Figure out how to hide the lines that were used in creating the vertices. (You want to display the polygon, not the underlying lines.)
• Create point H.
• Create the segmented path B-P-Q-S-H.
  • Create point B' by reflecting point B across line AC.
  • Create point B'' by reflecting point B' across line CD.
  • Create point B''' by reflecting point B'' across line DE.
  • Create line(HB''').
  • Create point S.
  • Create line(SB'').
  • Create point Q.
  • Create line(QB').
  • Create point P.
  • Create segments BP, PQ, QS, and SH, and give them those names.
  • Figure out how to hide the lines that were used in creating points P, Q, and S. (You want to display the segmented path, not the underlying lines.)
  • Also figure out how to hide points B', B'', and B'''.
• Get the length of segmented path B-P-Q-S-H to display as a quantity in your list of objects.
  • Click in the white box that is located at the bottom of the Geogebra window. A cursor will appear in the box. You are now in a mode where you can type a description of an object into this box.
  • Into the box type: path1=BP+PQ+QS+SH[enter]. Geogebra will respond by putting the quantity path1 in your list of dependent objects. The length of the path will be shown.
• Create another segmented path B-U-V-W-H and get the length of this path to display as a quantity in your list of objects.
  • Create points U, V, and W that are free to move on segment(AC), segment(CD) and segment(DE), respectively.
  • Create segments BU, UV, VW, and WH with those names.
  • Get the length of segmented path B-U-V-W-H to display as a quantity in your list of objects.
    • Into the box type: path2=BU+UV+VW+WH[enter]. Geogebra will respond by putting the quantity path2 in your list of dependent objects. The length of the path will be shown.

• Put the text "5.2.2.Lastname" somewhere in the drawing.
• Save the file with filename "5.2.2.Lastname" in the folder called Geometry.H5.Lastname that you created earlier.

Hints for 5.2#5a written solution

• Let point P have coordinates (0,h).
• Consider departing from point P along two different possible paths.
  • Path #1: This path starts by going down and to the right from point P, at an angle of 45 degrees with side(AB). Going along this path, you will soon hit side(AD). Call the intersection point V1. Note that the coordinates of point V1 are V1=(h,0).
  • Path #2: This path starts by going up and to the right from point P, at an angle of 45 degrees with side(AB). Going along this path, you will hit side(BC) at point Q, then hit side(AC) at point S, then hit side(CD) at point T, then hit side(BC) at point U, and then finally hit side AD. Call that final intersection point V2. Let’s consider how to determine the coordinates of point V2.
    • Segment(BP) has length BP = b - h.
    • Then BQ = b - h as well.
    • So point Q will have coordinates Q=(b-h, b).
    • So point S will have coordinates S=((b-h)+b,0). That is, S=(2b-h,0).
    • Proceeding in this way, you can get coordinates of points P, Q, S, T, U, and V2. Note that the coordinates of point V2 will be of the form V2=(some expression involving a and b and h, 0)
• But you want points V1 and V2 to be the same point. For this to happen, their x-coordinates must be equal. This gives you an equation some expression involving a and b and h = h.
• Eliminate h from the equation to end up with an equation involving a and b.
• Solve this equation for a in terms of b.

Hints for 5.2#10 written solution

• The wording of this problem is not as helpful. Your job is to create two chords, call them Chord1 and Chord2, with the following properties:
  • Chord1 goes through point A.
  • Chord2 goes through point B.
  • Chord1 is perpendicular to Chord2.
  • Chord1 has the same length as Chord2.
• Figure out how to create Chord2.
  • Imagine what happens to some points on Chord1 when Chord1 is rotated about the center of the circle by 90 degrees.
    • Let the endpoints of Chord1 be P and Q.
    • When Chord1 is rotated around the center of the circle by 90 degrees, the two points P and Q are rotated with it. The result is a new chord with endpoints P' and Q'.
    • Because rotation is an isometry, the distance PQ is the same as the distance P'Q'. So the new chord has the same length as Chord1. The new chord will also be perpendicular to Chord1. So the new chord must actually be Chord2. So, an easy way to get Chord2 would be to just rotate Chord1 by around the center of the circle by 90 degrees.
    • But we don't know what Chord1 is.
    • However, we do know that point A is one of the points on Chord1. If we rotate point A about the center of the circle by 90 degrees, the resulting new point A' must be one of the points on Chord2.
  • Do we know any other points on Chord2? I'm going to let you think about that one.
  • This gives you enough information to create Chord2.
• In an analogous way, figure out how to create Chord1.

Instructions for 5.3#3 Geogebra Drawing

• It has the same labeling as the figure at the top of page 275 in the book.
• Points A, B, C are free.
• Points A1, B1, C1 are dependent.
• Point A2 is free but is constrained to move along side(BC) in the manner described in the problem statement.
• Points C2, B2, A3, C3, B3, are dependent.
• The segmented path A1-C1-B1 is shown
• The length of segmented path A1-C1-B1 is displayed as a quantity in your list of objects.
• The segmented path A2-C2-B2-AC-C3-B3-A2 is shown
• The length of segmented path A2-C2-B2-AC-C3-B3-A2 is displayed as a quantity in your list of objects.
• The text 5.3.3.Lastname appears somewhere in the drawing.
• The file is save with the filename 5.3.3.Lastname in the folder called H5.Lastname that you created earlier.

• Hide objects that are used in the construction but do not appear in the picture in the book.
• Rename objects with descriptive names. For example, Geogebra may give segment(C3B3) the name k. You ought to change the name to C3B3.

Instructions for 5.3#4 Geogebra Drawing

• It has the same labeling as the figure in the middle of page 274 in the book.
• Points A, B, C are free and all other objects are dependent.
• Your list of objects includes segment(O1O2), segment(O2O3), and segment(O3O1), with names O1O2, O2O3, and O3O1. (Geogebra will display the lengths of these segments, and that will let us determine whether or not triangle(O1O2O3) is equilateral.
• The text 5.3.4.Lastname appears somewhere in the drawing.
• The file is save with the filename 5.3.4.Lastname in the folder called H5.Lastname that you created earlier.

• Hide objects that are used in the construction but do not appear in the picture in the book.
• Rename objects with descriptive names.

Written work Wrap-up.

Geogebra Wrap-up.

• You should now have a folder called Geometry.H5.Lastname on your computer desktop or on your thumbdrive. This folder should contain the three files called 5.2.2.Lastname, 5.3.3.Lastname, and 5.3.4.Lastname.
• Create a compressed (zipped version of this folder). On the lab machines, this can be done using the 7-Zip program. On other computers. On other computers, the procedure will vary.
• e-mail me the folder by the end of the day on Tuesday, February 17.
  • Use your O.U. e-mail account.
    • Recipients: yourself and me
    • Subject: "Geometry H5.Lastname"
    • Attachment: the compressed folder that you created.

Click here to go back to the web page for Winter 2009 Math 330A/539.

Click here to go back to Mark Barsamian's homepage.

Last updated February 11, 2009.