MATH 5200/7200, Fall 2008
Homework 8, due Wednesday 9/24
Correct your mistakes from Exam 1A (the in class part of Exam 1). Turn in your original exam together with your corrections.
Homework 9, due Friday 9/26
Prove the following theorems using our axioms and basic theorems. Include a GSP sketch with each of your proofs. The axioms and basic theorems are posted on the course web site. (You won't need the first four axioms for this assignment.) Definitions involving circles are also posted on the course web site.
1. Let P, Q, R, S be points on the circle C, with P not equal to Q and R not equal to S. Let A be the minor arc of P and Q, and let B be the minor arc of R and S. The arcs A and B have equal arc angle measures if and only if the chords PQ and RS have the same length.
2. Every trapezoid inscribed in a circle is isosceles.
3. The radius of a circle inscribed in an equilateral triangle is equal to one-half of the radius of the circumscribed circle and equal to one-third the altitude of the triangle.
Email your homework to Dr. McCrory by 9 am Friday morning.
Homework 10, due Monday 9/29
Prove the following theorems using our list of basic theorems and axioms. Use GSP to illustrate your proofs.
1. Let A, B, C, D be points on a circle c, with A not equal to B, and C not equal to D. Suppose that the lines AB and CD intersect at a point P. Then (PA)(PB) = (PC)(PD).
2. Suppose that a quadrilateral has a circle inscribed in it; in other words, each of the four sides of the quadrilateral is tangent to the circle. Then the sum of one pair of opposite sides is equal to the sum of the other pair of opposite sides.
3. If an equilateral polygon is inscribed in a circle, then it is a regular polygon.
Email your homework to Dr. McCrory by 9 am Monday morning.
Homework 11, due Monday 10/6
Proofs of the basic theorems from the axioms.
The class is divided into three groups as follows:
Group A: Brown, Chung, Dumont, Gold, Ivey, Newton, Wilson, Wright
Group B: Bonilla, Clark, Gibbs, Hallman, Lowe, Newman, Wallace, Whitt
Group C: Cho, Davenport, Hill, Juhasz, Newton, Mohl, Singletary, Slaughter
Each group will work on one or more sets of the basic theorems:
Group A: Set 1 - isosceles triangles (except the straight angle theorem), Set 2 - parallels (except the existence of parallels).
Group B: Set 3, congruence
Group C: Set 4 - area, Set 5 - similarity (except the side-angle-side similarity theorem)
The "rules of the game" are that the only facts you can use are (1) the axioms, or (2) the theorems in an earlier set. You may prove the theorems in each set in any order, but I recommend the given order. Once you have proved a theorem you can use it to prove other theorems in the set. If you need a fact that's not an axiom or in an earlier set, you have to prove it. (It is OK to check with the groups working on earlier sets to see if they've proved any extra facts you could use.)
I will prove the straight angle theorem, the existence of parallels, and the SAS similarity theorem in class. So in Set 1 you may assume the straight angle theorem, in Set 2 you may assume the existence of parallels, and in Set 5 you may assume the SAS similarity theorem.
All of class Wednesday 11/1 and part of class Friday 11/3 will be devoted to group work on this assignment.
Write-ups of the homework are to be emailed to Dr. McCrory by 9 am on Monday, October 6. I will call on undergraduates from each group to present the group's work on 11/6, 11/8, and possibly 11/10. Every undergraduate in a group must be prepared to present every theorem of the group!
Homework 12, due Monday 10/13
Prove the following theorems using only our axioms and basic theorems. Write your proofs in complete sentences, and be careful with your logic. Be sure to refer explicitly to the axioms and theorems you use. Illustrate your proofs with GSP sketches.
1.The line joining the midpoints of the nonparallel sides of a trapezoid is parallel to the parallel sides and equal in length to half their sum. (I assume the trapezoid is not a parallelogram. What happens if it is a parallelogram?)
2. The area of a trapezoid is 1/2 h (b1 + b2), where b1 and b2 are the lengths of the parallel sides and h is the length of a perpendicular from a point on one of the parallel sides to the line containing the other parallel side. (I assume the trapezoid is not a parallelogram. What happens if it is a parallelogram?)
3. Give reasons for each of the steps in the following proof outline. The reasons for each step can be one or more definitions, axioms, or theorems, together with one or more previous steps in the proof outline. (The steps have been organized into groups, and the conclusion of each group of steps is in boldface.)
Theorem. Let c1 be a circle and P a point outside c1. Consider two lines through P such that one line is tangent to c1 at A and the other line intersects c1 at B and C, with B between P and C. Then PA is the geometric mean of PB and PC.
Proof. Let O be the center of the circle c1.
1. Angle BCA = 1/2 angle BOA.
2. Angle BOA + angle ABO + angle OAB = 180.
3. OA = OB.
4. Angle ABO = angle OAB.
5. Angle BOA = 180 – 2 angle OAB.
6. 1/2 angle BOA = 90 – angle OAB.
7. Angle OAP = 90.
8. Angle BAP = 90 – angle OAB.
9. Angle BCA = angle BAP.
10. Angle PCA = angle BCA.
11. Angle PCA = angle BAP.
12. Triangle PAB and triangle PCA are similar.
13. PA/PB = PC/PA.
14. PA is the geometric mean of PB and PC.
Email your homework to Dr. McCrory by 9 am Monday morning.
Homework 13, due Friday, 10/17
Prove the following theorems using only our axioms and basic theorems. In problems 1, 2, and 3, write your proofs in complete sentences. Illustrate your proofs with GSP sketches.
1. Connecting the midpoints of the sides of a triangle divides the triangle into four little triangles. Prove that these little triangles are similar to the original triangle, and that all four of these little triangles are congruent.
2. Suppose that the circles c1 and c2 intersect at the points A and B, and AC is a diameter of c1, and AD is a diameter of c2. Prove that C, B, D are collinear.
3. Prove that for every quadrilateral the midpoints of the sides are the vertices of a parallelogram.
4. Give reasons for the numbered steps in the following proof.
Theorem. Let ABC be a triangle with altitudes AD, BE, CF. These altitudes are the angle bisectors of the triangle DEF.
Proof:
1. Triangle ABE is similar to triangle ACF.
2. (AB)(AF) = (AC)(AE).
3. Triangle AEF is similar to triangle ABC.
The same argument shows that triangle DBF is similar to triangle ABC, and triangle DEC is similar to triangle ABC.
4. Angle DFB = angle AFE.
5. Angle CFD = angle EFC.
6. Ray FC is the bisector of angle EFD.
The same argument shows that ray EB is the bisector of angle DEF, and ray DA is the bisector of angle FDE. This completes the proof.
Email your homework to Dr. McCrory by 9 am Friday morning.