Reading: Finish Reading ALL of Chapter 1 of Brown. We have now covered the whole chapter in class or lab, except that some parts of 1.5 have been left for you to work out outside of class.
Let ABC be an isosceles triangle (AB = AC). Suppose there is a point D on segment AB so that CD = CB. Let us label the lengths: AB = AC = x and CD = CB = y.
(a) With these assumptions what is length BD in terms of x and y? Also, write the length AD in terms of x and y.
(b) If we make the additional assumption that triangle DAC is also isosceles (i.e., DA=DC), find the ratio r = x/y. The answer should be a real number (no unknowns). Hint: If r = x/y, then x = ry, so you can substitute for x in your equation.
(c) Continuing with the assumption of (b), let angle BAC = a and let angle ABC = b. Find all the other angles in the figure in terms of a and b. Is it possible to derive a numerical value of a and b in this case?
Draw a rectangle ABCD with point E on AB and F on CD. Suppose that AEFD is a square and BCFE is similar to ABCD. Find the numerical ratio AB/BC.
· Make a model of a cube. Start with one vertex A and draw the 3 segments AB, AC, and AD that are all diagonals of square faces of the cube (the square faces must have A as a vertex).
· Then what shape is ABCD? Explain why.
· What are the remaining shapes?
· If you can assume that the volume of a pyramid is (1/3)*(base area)*(height), what are all the volumes of the pieces, including ABCD?
· Let H be a half-turn with center O (see Brown) and m be a line; let reflection in m be denoted by M. Draw O and m on a piece of graph paper.
· Draw a polygon S and construct the image MH(S). Explain carefully what isometry MH is. Give details (distance, direction, etc.)
· Also, explain what isometry HM is.
· Answer Brown, p. 20, #2.
· Answer Brown, p. 21, #4.
· Answer Brown, p. 22, #11. Do a nice construction.