January 1-10 - Cross Section of a Cube: A 'cross section' of a cube is a shape that you get when you cut the cube with a plane. Given a cube with a surface area of 96 cm^2, if you cut the cube with a plane that is parallel to one of its faces, you will get a square. What is the perimeter of that square? What is the perimeter of the largest rectangle you can get as a cross section? How can you get an equilateral triangle as a cross section? What are the areas of the square, rectangle, and the largest possible equilateral triangle?
January 13-17 - Half a Square & Semi-Circle: ABC is half a square inscribed in a semi-circle (A->B->C). Then a semi-circle is constructed on AB. BD is then constructed, the perpendicular bisector of AC, and triangle ABD is shaded, as is the part of the outer semi-circle that's not part of the original semicircle. What did Hippocrates of Chios prove about these two regions?
January 20-24 - Dividing up a Triangle: Take any triangle. Label it ABC. Construct D and E as the midpoints of BC and AB, respectively. Now construct DF and EF, where F is any point on AC. Now look at the triangles AEF and FCD and the quadrilateral BEFD. How are the areas of the triangles related to the area of the quadrilateral? What happens as F moves along AC? Can you explain why you think you're right?
January 27-31 - Rotated Goalposts: The Superbowl goalposts are 18'6" wide. The uprights are 30' tall and are attached to a post at the middle of the crossbar. If the uprights are accidentally bent down and rotated around the middle post so the corner of the goalposts winds up 1 inch lower than where it started, how far is the top of one of the uprights now from where it started?
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