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Private: Learning Math: Geometry
Dissections and Proof Part B: Cutting Up (50 minutes)
Session 5, Part B
You can solve each of the following problems in more than one way. Try to find several solutions; then pick the one that you like best. Jot down notes about each solution so that you can reconstruct your work later. Print out these figures to use in Problems B1-B6.
For each problem, start with the figure described. Find a way to cut that figure into pieces you can rearrange to form the second figure described. You have to use all the pieces from the original figure and put them together like a puzzle — no gaps or overlaps are allowed. Pay attention to how you can justify your cutting process. Use the properties of your beginning figure and the cuts you made to explain why you ended up with the second figure. Note 2
Problem B1
Start with a parallelogram. Find a way to cut your parallelogram into pieces you can rearrange to form a rectangle.
Problem B2
Start with a right triangle. Dissect the triangle so that you can rearrange the pieces to form a rectangle.
Problem B3
Start with a scalene, non-right triangle. Cut it into pieces that will form a parallelogram.
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Video SegmentIn this video segment, groups of participants work to transform several shapes into new figures. As they progress toward more difficult problems, they are able to make increasingly complex justifications for why their cuts work. Were you able to make similar kinds of justifications? You can find the first segment on the session video approximately 5 minutes and 45 seconds after the Annenberg Media logo. The second segment begins approximately 8 minutes and 52 seconds after the Annenberg Media logo. |
If you’ve solved Problems B1 and B3, you can put those solutions together to solve Problem B4. Alternately, you can divide the triangle into two right triangles and apply your solution to Problem B2 twice.
Problem B4
Start with a scalene, non-right triangle. Cut it into pieces that will form a rectangle.
Note 3
Take It Further
Problem B5
Start with a trapezoid. Dissect the trapezoid into pieces that will form a rectangle.
Problem B6
Start with a trapezoid. Dissect the trapezoid into pieces that will form a triangle. As a challenge, see if you can do this with a single cut.
Problem B7
In several of the problems above, you had a rectangle as a final result. For each of the following shapes, start with a rectangle and dissect it into pieces you can rearrange to form that shape:
- An isosceles triangle
- A right triangle
- A scalene triangle
- A non-rectangular parallelogram
- A trapezoid
Problems B1-B5 and the Video Segment problems adapted from Connected Geometry, developed by Educational Development Center, Inc. p. 164. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math
Notes
Note 2
If you are working in a group, pose the first problem and give everyone a couple of minutes to work on it. Watch for someone with an appropriate solution. Ask that person to share the solution and explain why it works. That’s the goal for these problems: to come up with methods that will always work and that don’t rely on measurement. Also, make sure that everyone is clear about the “rules of the game,” particularly that you have to use all the pieces and you have to be able to describe your method using the language of flip, rotate, and translate. At the end, leave at least 10 minutes to share methods that can be explained, as opposed to those that just look right.
Note 3
After you have solved Problems B1-B4, reflect on why your cuts work. It’s likely that you used something like a “midline cut” to turn a triangle into a parallelogram and then into a rectangle. Consider that we want a single cut to do two different things: connect consecutive midpoints (so that the segments you attach will match up), and form a segment that is parallel to the base (so that the final figure will have parallel sides). This leads into Part C, where we prove that these two segments really are one and the same.
Solutions
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).
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| 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.)
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| 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.
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| 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.
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