Math 444 Autumn 2001

Assignment Due Friday 10/19

Computational Problems with Centers of Mass and Barycenters

  1. Let A = (1, 7), B = (4, -2).  Find the center of mass of a system with mass 2 at A and mass 1 at B.
  2. If C = (0,1), find the center of mass of a system with the same masses at A and B, and a mass 3 at C.
  3. Suppose P, Q, R are points, with specific coordinates not specified.  If masses 2, 1, 3 are placed at P, Q, R, find the center of mass M of the two masses at P and Q.  Then find the center of mass of a system with mass 3 at R and also mass 3 = 2 + 1 at M.
  4. Given 4 points A, B, C, D, with mass =1 at each point.  Compute the midpoints M of AB and N of CD and then the midpoint E of MN.  Then compute the midpoint P of AC and Q of BD and then the midpoint F of PQ.  Finally, compute the midpoint R of AD and S of BC and then the midpoint G of R and S. How are the points E and F and G related?
  5. Prove with algebra that for any quadrilateral ABCD, the quadrilateral of midpoints of sides of ABCD is a parallelogram.  (You can assume that a quadrilateral is a parallelogram if and only if the diagonals bisect each other.)

Beginning Congruence Theorems

  1. A quadrilateral ABCD is called a kite if AB = AC and also DB = DC.  For such a quadrilateral, the diagonal AD is the perpendicular bisector of BC.  Also prove that angle ABD = angle ACD. (Note:  You can use any of the 3 congruence criteria SSS, SAS, ASA from lecture.  Also consider the proof of the result about isosceles triangles.)