Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

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