Assignment 4 (Due Wed. 10/20)

Problem 4.1

• Write the proof in BG problem 3.3.
• Construct a square ABCD.  Call the side length s.
• Construct a rectangle EFGH whose area equals area ABCD but with EF = s*sqrt 2.
• Construct a rectangle IJKL whose area equals area ABCD but with IJ = s * sqrt 5.

Problem 4.2

Do BG 3.15.  Make a construction or careful drawing in each case of a decomposition of both figures into pieces so that the pieces for first figure can be can be reassembled to form the second figure.

Problem 4.3

Do BG problem 3.7. 

Problem 4.4

This figure is a map of adjoining farms.  Thales owns the first farm, with boundary ABFE, and Euclid owns the second farm, with boundary CDEFB.  The two farmers wish to replace their common boundary EFB with a new straight boundary line. However, they want the new farms to have the same areas as the old ones.  Construct a line segment connecting AC and AD that can serve as the new boundary.  Explain and justify your construction. (You can draw your own figure so long as it is approximately like this one.  It should have no special properties of distance, angle, etc.)

Problem 4.5

This figure is a "strip" figure as found in Lab 2.  Specifically each dashed line is a line parallel to one of the sides of triangle ABC and whose distance from this side equals the length of d.  For example, line A1B1 is parallel to line AB.

• If we denote the side lengths AB = c, BC = a, and CA = b, what is the area of the region contained inside triangle A2B2C2 and outside triangle A1B1C1?  Explain your reasoning.

• Is A on line A1A2? What special properties does line A1A2 possess?  Does it have a special name?  Prove your assertions. (You can use quote results from assignment 3 and earlier in your proof.)

• Prove that lines A1A2, B1B2, C1C2 are concurrent at some point P. 

• Explain how to use the point P above to construct a circle of radius r inscribed in triangle ABC.

• Using only a, b, c and r, find a formula for the area of triangle ABC.  (This is a simple formula.  It is not Heron's formula.  It has no square roots.  Hint: what is the area of triangle PAB?)