Solutions for Session 5, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6| B7

 Problem B1 Draw a perpendicular DE. Cut along the perpendicular to form a right triangle. Then translate the right triangle to the right until the side AD coincides with the side BC. Note you could draw a perpendicular line anywhere along the side of the parallelogram (not just the vertex) as long as the perpendicular lies within the parallelogram. It is always possible to find one perpendicular line within the parallelogram between at least one pair of parallel sides.

 Problem B2 Construct the midpoints of the sides BC and AC, namely D and E, respectively. Connect the two midpoints. Cut along the segment ED. Then rotate the triangle EDC 180° about the vertex E. Notice the sides EC and AE will coincide.

 Problem B3 Proceed exactly as in Problem B2 by constructing a midline and rotating the top triangle 180° about the vertex D. The result will be a parallelogram instead of a rectangle.

 Problem B4 Start with a scalene, non-right triangle. Use the method of Problem B3 to form a parallelogram. Then apply the method of Problem B1 to get a rectangle from the parallelogram. Note that sometimes you may need to reposition the parallelogram before you turn it into a rectangle.

 Problem B5 Connect the midpoints G and H of the sides AD and BC, respectively, with a line segment GH. Cut along GH and rotate the trapezoid DCHG about the point G, counterclockwise, until the segments GD and AG overlap. The resulting figure will be a parallelogram. Then apply Problem B1 to create a rectangle.

 Problem B6 Starting with a trapezoid, you can use the process in Problem B5 to make a rectangle. Cut the rectangle into halves along its longer side (or shorter side, which works just as well). Then cut one of the smaller rectangles into two triangles by drawing a diagonal (see picture). Rotate triangle ADF clockwise about the vertex F until the sides DF and FC overlap. The resulting figure is a triangle. Challenge: Position the trapezoid so the parallel sides are horizontal, with the shorter one on top. Connect the top right vertex to the midpoint of the left (non-parallel) side. Cut along this segment to form a triangle on top and a quadrilateral. Rotate the triangle 180° about the midpoint of the left side. You now have a triangle with base = (sum of two bases of the trapezoid) and height that is the same as the height of the trapezoid. The attached sides match up because you cut at a midpoint. The bottom side is straight because the bases are parallel in a trapezoid, so adjacent angles (bottom and top) are supplementary.

Problem B7

 a. Cut the rectangle along a diagonal, creating two triangles. By translating and flipping one of the triangles, put them together into one triangle whose congruent sides correspond to the diagonal of the original rectangle. b. Start with the rectangle ABCD. Find the midpoint, E, of one of its sides. Connect E with the vertex B, and cut along the segment EB. Rotate the triangle EBC counterclockwise until the sides ED and CE overlap. The resulting triangle is a right triangle (see picture). c. Start with the rectangle ABCD. Mark an arbitrary point E (not the midpoint!) along the top side CD. Then find point M and N such that M is the midpoint of CE and N is the midpoint of DE. Cut along segment BM, and rotate triangle BCM 180° about point M. Segments EM and CM will coincide. Now cut along segment AN, and rotate triangle ADN 180° about point N. Segments DN and EN will coincide. Segments AD and BC will also coincide. (It's possible that one of the resulting sides will be the same as the side of the rectangle, but for all but three choices of point E -- the excluded midpoint being one of the choices -- this will produce a scalene triangle.) d. Start with the rectangle ABCD. Choose any point E on the segment CD (other than C and D themselves). Connect E and B with a line segment. Cut along the segment EB. Translate the triangle EBC so that the sides BC and AD overlap. The resulting figure is a parallelogram but not a rectangle. e. Use the solution from Problem B7 (d) to create a parallelogram. Take the shaded triangle and reflect it about a line parallel to the horizontal sides of the parallelogram, cutting it in half.