You will receive a document called Getting Acquainted with Sketchpad. Work through the exercises with Sketchpad. Make note of your answers to the questions about quadrilaterals on the last page of the first section; you will want to refer to them later.
Do these exercises before coming to class.
(1) You were given a printed triangle ABC and assigned a letter A, B or C. You are also given masses of 1 at A, 2 at B, and 3 at C. Depending on which letter you were assigned, draw very carefully the following points in your triangle.
|
Letter Assigned |
Task |
|
A |
Find the location A’ of the center of the masses at B and C. Then find the location A’’of the center of mass of the mass at A and the mass 2+3 =5 at A’. |
|
B |
Find the location B’ of the center of the masses at C and A. Then find the location B’’ of the center of mass of the mass at B and the mass 3+1=4 at B’. |
|
C |
Find the location C’ of the center of the masses at A and B. Then find the location C’’of the center of mass of the mass at C and the mass 1+2=3 at C’. |
(2) Let two (non-vertical) planes U and V be given in (x,y,z) space. The vertical projection of one plane to another is defined at follows. Let P be a point on U. Then let p be the vertical line through P (i.e., the line parallel to the z-axis). Then line p intersects plane V in a point P’. P’ is the vertical projection of P. This defines a rule, or a function from the plane U to the plane V. Question: If A = (1,0,0), B = (0,1,0) and C = (0,0,1), let U be the plane containing A, B, C. If V is the (x,y,0) plane, what is the projection A’B’C’ of ABC on V? What kind of triangle is ABC? What kind of triangle is A’B’C’?
(3) Suppose you are given any rectangle ABCD. Explain a way to position the rectangle in 3-space so that the rectangle is projected to a square. Write down your explanation so that it can be read in class.
(4) Perform this experiment. Cut out a triangle ABC. Now find the corner of a box and drop the triangle into the corner so that A lies on one edge, B on one edge and C on one edge. Then what kind of triangle is the projection of ABC onto one face of the box? Whatever the triangle, can you always drop it into the corner of a box like this, or will some triangles not fit?