Assignment 2, Due Wed 1/15

2-1 Constructing the plane projection of a quadrilateral using diagonals

In the table below, you are given the coordinates on centimeter graph paper of the vertices of a quadrilateral ABCD and also the lengths of segments AA', BB', CC''.  The task is to construct a model somewhat like the model of Assignment 1.  The key part of the model will be a cardboard right prism Q with base ABCD and top a plane quadrilateral A'B'C'D' with the given lengths AA', BB', CC'.  This means that you will need to construct DD' and some other lengths. You can use a ruler and/or graph paper to help in your construction, but you should also explain how to compute the needed quantities.

A = (0,10); AA' = 6 cm

B = (0,0); BB' = 12 cm

C = (8,2); CC' = 20 cm

D = (6, 12); DD' = ??

Now that you have some practice making such a model, we will firm up the rules so that the geometry needed comes to the fore.  This time you should construct the sides before assembling the model.  You should not just eyeball to find a reasonable DD'.

Let E be the intersection of diagonals AC and BD. Construct the following pieces before assembling the model and give a clear write-up explaining how they were constructed.

• Construct the diagonal piece ACC'A' and on this piece construct EE'.
• Then construct the diagonal piece BDD'B'.  The segment EE' is on this piece also, and you will need it to construct DD'.
• Construct the faces ABB'A', BCC'B',CDD'C', DAA'D'.
• Cut slits in the diagonal pieces along EE' so that they interlock.  Assemble the whole model with diagonals and the 4 faces.  The diagonals should keep it rigid.

2-2 Constructing the plane projection of a quadrilateral using parallelograms

This will be a second model that can fit inside the same cardboard right prism Q.

• Construct points D1 on BC and D2 on BA so that BD1DD2 is a parallelogram.
• Then construct a prism P with base BD1DD2 so that the top points B'D1'D'D2' are on the same plane as A'B'C'D'.
• If you remove the diagonals and place P inside Q the top points should all lie in the plane.

Explain your construction.  In particular explain why B'D1'D'D2' is also a parallelogram and how the vertical lengths were constructed from the lengths that were already known.

2-3 Dear Diary:

Write a thoughtful paragraph reflecting on the plusses and minuses of the model-building problems as methods for learning geometry. 

• What did you learn? Did you learn some mathematics from confronting construction issues that you might have missed otherwise? 
• Did you like this better or worse than problems of a more conventional nature?  Should this have been a group problem?  Do you think that a problem of this nature would be appropriate in a high school classroom?

2-4 Find the lengths in the triangle

In this figure, the lines through D are parallel to the sides of triangle ABC.  Suppose the side lengths of ABC are a = BC, b = CA, c = AB.  Also suppose that x = BF3/BA, y = CF1/CB, z = AF2/AC.

(a)    Find the lengths of all segments in the figure.  Make a neat table. Justify your answers, being complete without being repetitive.

(b)   Find all the areas of the 3 quadrilaterals and 3 triangles inside ABC, if we assume that area ABC = K. Justify your answers, being complete without being repetitive.

(c)    The line AD intersects line BD in point A'.  What is A'B/A/C?

(d)   Find the center of a dilation that dilates ABC to DE1F1.  What is the ratio of similitude?

2-5 Algebraic Exercises

Let A = (2,71), B = (33, -5), C = (30,200), D = (3, 2).

Let  E = (1, 3, 5), F = (3, 1, 4), G = (30, 13, 2), H = (1, 1, 1).

(a)    Compute the following points:

• Midpoint M of AB and midpoint N of CD.  Midpoint of MN.
• Midpoint O of AC and midpoint P of BC.  Midpoint of OP.
• Centroid U of ABC.

(b)   If an affine map S maps (0,0) to A, (1,0) to C and (0,1) to D, what is S(2,3)?  What is S(x,y)?

(c)    Compute the following points:

• Midpoint X of EF and midpoint Y of GH.  Midpoint of XY.
• Midpoint Z of EG and midpoint W of FH.  Midpoint of ZW.
• Centroid U of EFG.

(d)   If an affine map T maps (0,0) to E, (1,0) to F and (0,1) to G, what is T(2,3)?  What is T(x,y)?