MATH 5200/7200, Fall 2008

Homework 1, due Wednesday, August 20

1. Use Geometer's Sketchpad (GSP) to construct the four triangle centers discussed in class: the orthocenter, the incenter, the centroid, and the circumcenter.

2. Think of some questions coming from these constructions. For example, when are the four centers the same? Why are the three altitudes of a triangle concurrent? And discuss the answers to some of your questions!

Submit the homework by email to Dr. McCrory, preferably by 9 am Wednesday. For problem 1 submit your GSP files.

Homework 2, due Friday, August 22

1. (a) Use Geometer's Sketchpad to discover the relation between the angles of a triangle and the arc angles of the circumcircle. In other words, for triangle ABC, find the relation between:

- the angle A and the arc angle BC of the circumcircle,
- the angle B and the arc angle CA of the circumcircle,
- the angle C and the arc angle AB of the circumcircle.

The relation should hold for all three angles of all triangles, acute or obtuse. Illustrate your conclusion with a GSP sketch and measurements.

(b) (Extra Credit) Prove this relation.

2. (Extra Credit) Given a triangle ABC, how do you find the point O such that the sum OA + OB + OC is minimal?

Submit the homework by email to Dr. McCrory by 9 am Friday.

Homework 3, due Monday, August 25

Do an internet or textbook search for proofs of the Pythagorean Theorem, pick your favorite proof, and write it up to present to the class. Don't just cut and paste things from the web -- make your own GSP sketches to illustrate the proof. You can write the text of the proof on your GSP file. No matter how you choose to present the text, please be careful to make it large enough so it can be read on the computer projection screen.

Give a reference (internet address or textbook title and author) for your proof.

I will call on several people to present their proofs to the class Monday. (I'll bring everyone's files to class on a flash drive.)

Submit the homework by email to Dr. McCrory by 9 am Monday.

Homework 4, due Friday, August 29

Do the following constructions using only the GSP Construct menu. Do not use the Transform or Measure menus to do your constructions. This is the GSP version of the ancient Greek problem of construction with ruler and compass. Use the Measure menu to check that your constructions are correct.

1. Construct the following regular polygons, starting from a side of the polygon: equilateral triangle, square, regular hexagon, regular octagon. (A regular polygon has all sides equal and all angles equal.)

2. Given a square S, construct another square S' such that the area of S' is twice the area of S.

3. Let x and y be positive real numbers. The geometric mean of x and y is the number m such that m² = xy. In other words, m = √(xy).

(a) Let ABC be a right triangle with right angle at C. Let CD be the altitude from the vertex C to the hypotenuse AB. Using similar triangles, show that CD is the geometric mean of AD and BD.

(b) Given two segments of length x and y, construct a segment whose length is the geometric mean m of x and y. Hint: If a triangle ABC is inscribed in a circle, and AB is a diameter of the circle, then C is a right angle.

(c) Given a rectangle, construct a square with the same area.

Submit the homework by email to Dr. McCrory by 9 am Friday.

Homework 5, due Wednesday 9/3

In these construction problems use only the Greek construction rules and the GSP construction rules which can be derived from them, i.e. all of the GSP Construct menu except Point on Object and Locus. The solution to each problem should include a GSP sketch of the construction (in electronic form) and a proof that the construction works.

1. (a) Given points A and B, show how to construct two points C and D which divide the line segment AB into 3 segments of equal length. Hint: Use properties of parallel lines.

(b) Explain how to do a construction which divides the line segment AB into n segments of equal length (n > 1). Illustrate your construction for the case n = 7.

2. Given a triangle ABC, construct a similar triangle A'B'C' so that the area of A'B'C' is twice the area of ABC.

3. Given a circle C and a point P outside C, construct the two lines through P which are tangent to C. Definition of tangent: Suppose that C is a circle with center O, and A is an intersection point of C and a line L. The line L is tangent to C at A if the radius OA of C is perpendicular to L.

Submit the homework by email to Dr. McCrory by 9 am Wednesday.

Homework 6, due Monday, September 8 - Friday, September 12

Group presentations.

Homework 7, due Monday 9/15

Come up with at least one question about the student presentations this week. Be as specific as you can -- Don't just say "I didn't understand what X's group did." Refer to the files posted on the course web site for each group. We will discuss these questions in class Monday.

Email your question or questions to Dr. McCrory by 9 am Monday.