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Geometry - Wrecked Angle Paradox Worksheet

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                               Wrecked Angle Paradox
Credits
Adapted from Wrecked Angle from John Handley High School Mathematics Department,
http:/ / www.pen.k12.va.us/ Div/ Winchester/ jhhs/ math/ mhome.html.

Construction
Cut out the shapes below. Make as many shapes as you can. Record the apparent size
of the shapes by measuring the outside lengths and calculating the area.




                                   A

                                                                             B



                   C




                                              D




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Wrecked Angle Paradox




                        THI S PAGE I NTENTI ONALLY LEFT BLANK




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                                                                            Wrecked Angle Paradox



Paradox
1. Determine the area of each of the
   four original shapes.                                    A. Square:           _____________

    A. Right Triangle:         __________                   B. Rectangle: __________

    B. Right Triangle:         __________                   C. Triangle: ______________

    C. Trapezoid:              __________                   D. Trapezoid: _______________

    D. Trapezoid:              __________                   E. Octagon: _______________

                                                            F. Parallelogram: _____________
2. Determine the area of each of the
   figures that you formed with the                         G. Another Triangle:_____________
   four shapes:

 There appears to be a paradox! Some of the figures have 63 square units, some have
64 square units, and some have 65 square units. This seems paradoxical since all the
figures were formed with the same four pieces!

Here are some solutions for the figures above:


                       D                A                       B
              A                                 C
                                   B
                                        D
                   B   C
                                                            A       C   D




               B
                                                        D
                           A                    B
                                                                                A D
                       C
                               D
                                            A   C
                                                                            B         C

                               C
                                                A                       D
                                                        B       C
                                   D
               A
                                                    A
                   B                                        B
                                                                    C       D




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Wrecked Angle Paradox


Analysis
3. How do you explain this impossible situation? USE COMPLETE SENTENCES. See
Revealed Wrecked Angle,
http:/ / www.metacafe.com/watch/ 334953/revealed_wrecked_angle_revealed/ .




Conclusions
Get the answer key from the teacher and check your answers. Then answer the
following.


4. A visual proof is a proof that depends upon the accuracy of a drawing. What can
you conclude about the reliability of visual proofs? USE FULL SENTENCES




Other Resources
ƒ John Handley High School Mathematics Department, Wrecked Angle, 2006,
  http:/ / www.pen.k12.va.us/ Div/ Winchester/ jhhs/ math/ puzzles/ wreck.html, last
  accessed 3/3/2007.

ƒ Ashrak, Metacafe, Revealed Wrecked Angle,
  http:/ / www.metacafe.com/watch/ 334953/revealed_wrecked_angle_revealed/, last
  accessed 3/3/2007.

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                                              Answers



Answers to Analysis

Take a closer look at the first two figures -- the square and the rectangle. The four
pieces fit snugly together for the square.

The area of the square is 64 square units, which makes sense since the areas of the
four smaller pieces are 12 square units, 12 square units, 20 square units, and 20 square
units.

The area of the rectangle is 65 square units, but where does that extra square unit
come from?

Look at the diagram below and you will notice that the pieces do not fit snugly together
in the rectangle (the extra square inch can be seen in the middle).

How can you prove this?

If the pieces indeed did fit together to form a rectangle, then the diagonal would be
one line segment. If it is just one segment, then it would have just one slope, since the
slope of a line is always the same. Determine the slopes of the four pieces that form
"the" diagonal (the slopes should be 2/5, 2/5, 3/8, and 3/8). Notice that they are not
the same. Therefore, it is not one segment (which can be seen in the drawing above).




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