Assignments for Week 3

Reading for Week 3

B&B, Chapter 4.

This chapter contains the basic properties of parallel lines and transversals. It also introduces coordinates and the slope of a line.

Heilbron reading to be announced later.

Exercises 5 (Due Wed 10/13)

E5.1.Midpoint Quadrilaterals
Given a quadrilateral ABCD, the midpoints of the sides are the vertices of a quadrilateral which we call the midpoint quadrilateral of ABCD. Prove that the midpoint quadrilateral of ABCD is a parallelogram.
E5.2. Constructing parallels
Draw a line AB and a point C not on the line. Construct with straightedge and compass the line through C which is parallel to line AB.

Exercises 6 (Due Fri 10/15)

E5.1. Parallogram Diagonals
State the definition of a parallelogram. Prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other. (In other words, the diagonals intersect at a point M which is the midpoint of each diagonal.)
E5.2 A Construction of a parallelogram
Given 3 noncollinear points A, B, C. Use E5.1 to construct a point D so that ABCD is a parallogram.

Assignment 3 (60 points, Due Mon 10/18)

In these proofs you can use the theorems in Chapters 1, 2, 3 of B&B. Also, for problems from Chapter 4 of B&B you can use theorems that precede in the book what you are trying to prove. You can also use what you yourself prove in the homework so long as you keep a logical sequence and do not end up with circular reasoning.

3.1 Midpoint Quadrilaterals (10 points)

(a) For a quadrilateral ABCD, prove that the midpoint quadrilateral of ABCD is a parallelogram. (Yes, this is a repeated of an earlier exercise.)

(b) If ABCD is a kite, what kind of polygon is the midpoint quadrilateral? Prove your assertion.

(Note: In a problem of this kind you should give the best or strongest possible answer whether or not it is spelled out each time. For example, it is true that the polygon is a parallelogram, but this is not a satisfactory answer.)

3.2 Parallelograms (15 points)

Prove the following problems on B&B, pp 113-114. Important: Use the definition of a parallelogram at the beginning of the Exercises on p. 113. Do not use other equivalent definitions.

For tools, you can use anything proved in B&B through p. 122.

  1. Prove 2 and its converse 5.
  2. Prove 3 and its converse 7.
  3. Prove 6.

3.3 Rhombi (10 Points)

(a) A rhombus is defined to be a quadrilateral with four equal sides. Prove that a rhombus is a parallelogram.

(b) Prove that a parallelogram with perpendicular diagonals is a rhombus.

3.4 A special triangle (10 points)

Let ABC be an isosceles triangle (AB = AC = x) and let D be a point on segment AB so that CD = CB = y.

(a) What are lengths BD and AD in terms of x and y?

(b) If triangle DAC is also isosceles (i.e., DA=DC), find the the ratio x/y.

3.5 Special Midpoint Quadrilaterals (15 points--or more for unusually good answers)

(a) Consider this statement: If the midpoint quadrilateral of quadrilateral ABCD is a rectangle, then ABCD is a kite.

Prove it if it is true. If it is false, give a counterexample and, if you can, explain for which quadrilaterals the midpoint quadrilateral is a rectangle.

(b) Consider this statement: If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle.

Prove it if it is true. If it is false, give a counterexample and, if you can explain for which quadrilaterals the midpoint quadrilateral is a rhombus.

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