Assignment 4C

Assignment 4C (Due Wednesday 10/29)

Reading: Read B&B Chapter 5. Problems from this chapter will be done mostly in class.

Note: The midterm is Friday, 10/31. You may want to make a copy of your work to study from, since this will not be returned before the test.

4.3 Midpoint Quadrilaterals (6 points)

If ABCD is a quadrilateral, then its midpoint quadrilateral MNOP is the quadrilateral with vertices at the midpoints of the sides: M = midpoint AB, N = midpoint BC, etc.

For a quadrilateral ABCD, prove that the midpoint quadrilateral of ABCD is a parallelogram.

4.4 Special Midpoint Quadrilaterals (10 points)

Prove, if true. Construct a counterexample if false.

If ABCD is a kite, then its midpoint quadrilateral is a rectangle.
If the midpoint quadrilateral of ABCD is a rectangle, then ABCD is a kite.
If ABCD is a rectangle, then its midpoint quadrilateral is a rhombus.
If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle.

4.5 Median Intersection (10 points)

Draw a triangle ABC and construct midpoints D, E, F as in the figure. Let G be the intersection of the two medians CF and BD. Then let H be the midpoint of BG and I be the midpoint of CG.

Prove that DFHI is a parallelogram.
Then use what you know about parallelograms to find the ratio BG/BD and also CG/CF.
Finally, repeat this construction for medians BD and AE, intersecting in J. Can you prove G = J?

4.6 Two-edged ruler figure (6 points)

Suppose you place your ruler on paper and without moving it, draw lines along both edges of the ruler. Then turn the ruler (no special angle) and draw lines along both edges again.

Prove that the quadrilateral formed by the 4 lines is a rhombus.

More formally, there are given two parallel lines m1 and m2 which are distance d apart. Let n1 and n2 be another pair of parallel lines that are the same distance d apart, but which are not parallel to the first pair. Then the lines will intersect at four points, forming a parallelogram (by definition!). Prove that this parallelogram is a rhombus.

4.7 Circumcenter of an Isosceles Triangle (8 points)

Given an isosceles triangle ABC, with |AB| = |AC| = b and |BC| = a, what is the radius R of the circumcircle of ABC? (The answer should be in terms of a and b, if possible.)

Return to Math 444 Home Page.