Homework Assignments

Math 330 Fall 2001

Homework 1: due September 14

Section 4.1 #21. Also, what happens if the triangle is isosceles?
Section 4.1 #26

Section 4.2#6
Section 4.2#9
Section 4.2#12

Homework 2: due September 21

Section 4.3 #6, 7, 8, 15(a), 16 (figure only..no proof needed).

 

Homework 3: due September 26

Section 4.4

•  #8
• #14 (No proof needed. Make a good sketch with Geometer's Sketchpad and print it out. Include a paragraph explaining what you did in the sketch, and how it "verifies" the assertion.)
• #15 (Use a Geometer's Sketchpad printout. Include comments!)

 

Homework 4: due October 5
 For construction problems, you may use Geometer's Sketchpad. The rule is that you can use only the following constructions:

• circle tool

• line/segment/ray tool

• point on object

• point at intersection

You may write your own scripts for primitive constructions shown in appendix 6 and use them. 

In each case print out a script and a sketch. Remember, if you have a script directory set, then to open a script for printing, you need to hold the shift key down while you select the script. 

• Section 5.1 #7, 8
• Given a point A on a circle c, construct the line tangent to c at A.

Homework 5: due October 15

The rule is that you can use all constructions under CONSTRUCT.
In each problem, turn in a sketch and a script.

• Section 5.2 #20. 
• Section 5.3 #8, 12, 15, 20

Homework 6: due October 26

• Section 6.1 #23, 24  
• Section 6.2 #23 (Do not hide anything...show auxiliary objects with LINE STYLE DASHED and include a support script).
• Section 6.2 #15

Homework 7: due November 2

• In class you were given a copy of pages 346-355 from the book Geometry: Euclid and Beyond, by Robin Hartshorne. (Springer Verlag, 2000). The chapter copied is titled "Digression: Circles Determined by Three Conditions." Your assignment is to do construction 38.7 (The problem of Apollonius) with geometer's sketchpad. It calls for the construction of a circle tangent to three given circles. For inversions, be sure to use constructions and not transformations.

Homework 8: due November 9

1. Refer to Figure 6.22 page 265. Given triangle ABC, construct the incircle and one excircle. Then as in the proof, construct the line JH.  Construct the other two lines corresponding to the other two excircles.
   
    Do not hide anything. Since this is a construction problem, you cannot use the script we wrote for inversion, because it uses transformations.
   
   Do you see any relationships between triangle ABC and the triangle formed by the three lines that you constructed?
   
2. Consider the ellipse x^2+(y/2)^2=1. Use graphing methods from the class on Friday November 3 to represent this ellipse in Geometer's Sketchpad.  The ellipse should have vertices at (-1,0), (1,0), (0,-2) and (0,2).
   
   Illustrate the inverse of this ellipse through the circle centered at the origin with radius 3. Use algebra as in the second example on page 257 to find an equation for the image curve.
   
3. Consider the parabola (x+1/2)=(1/2)*y^2. The parabola opens to the right, has vertex at (-1/2,0),  and has y-intercepts +1 and -1.    Illustrate the inverse of this parabola through the  circle with radius 2 centered at the origin.  Use algebra to find an equation for the image curve.
   
4. Consider the hyperbola xy=1. Invert this through the unit circle. Use algebra to find an equation for the image curve.
   

Homework 9: due November 21

1. Do exercises 7.5#2-4.
2. Do exercises 7.5#5-7.
3. Do exercises 7.5#14-15.
4. Do exercises 7.5#16-17.

Homework 10: due December 5

1. Do exercises7.3 #13, 22 (use Hide/Show instead of overlays. Submit your work as a gsp file in W Math 330 HW10 folder. PLEASE INCLUDE YOUR NAME IN THE FILE.)
2. Do Exercise 1.6#2, 4, 5 (Really...no typo here.)
3. Do Exercise 1.6#15, 16, 20.

David Hilbert, one of the greatest geometer's of all times, says the following:

"Any theorem concerned solely with incidence relations in the plane can be derived from the theorems of Desargues and Pascal. And we have now seen that Desargues' Theorem is a consequence of Pascal's. Therefore we may say that Pascal's Theorem is the only significant theorem on incidence in the plane and that this configuration thus represents the most important figure in plane geometry. "

Wow!

Homework 11: due December 14

1. Given five coplanar points, no three of which are collinear, there is a conic passing through the five points. In writing, explain how to construct a general point on the conic.
2. Explain how to construct the general point in the configuration above if one of the five given points is at infinity.
3. Explain the dual construction for the question above (#2). So you are given five lines, but one of the lines is the line at infinity.