Question 1163283
{{{drawing(400,4800/13,-1,12,-1,11,

line(0,0,0,8),line(0,8,8,8),line(8,0,8,8),line(0,0,8,0),
line(3,10,11,10),line(11,2,11,10),line(3,2,11,2),line(3,10,3,2),
line(0,8,3,10),line(8,8,11,10),line(0,0,3,2),line(11,2,8,0),

green(line(3,10,1.5,1),line(1.5,1,8,0),line(9.5,9,8,0),line(3,10,9.5,9)),

locate(0.05,8,A),locate(0,0,B),locate(8,0,C),locate(8.05,8,D),
locate(3.05,10,E),locate(3,2,F),locate(11,2,G),locate(11.05,10,H),
locate(9.5,9,I),locate(1.5,1,J), locate(3.9,0,2), locate(11.1,6,2),
locate(9.6,1,2)


)}}}

The figure ABOVE shows a 2 × 2 × 2 cube ABCDEFGH, as well as midpoints I and
J of its edges DH and BF. It so happens that C , I , E , and J all lie in a
plane. Can you justify this statement?<pre>Yes I can. A quadrilateral is coplanar if and only if the sum of its
interior angles is 360°.  The plane of square ABFE is perpendicular to the
plane of BCGF. If two planes are perpendicular, then any angle whose sides
are in the two planes are 90°, so ∠CJE=90°  Thus EJ ∥ CJ and
similarly the other 4 interior angles of quadrilateral CIEJ also 90°. Thus
quadrilateral CIEF is also a rectangle.  Thus the sum of the interior angles is 
4∙90° = 360°.</pre>What kind of figure is quadrilateral CIEJ,<pre>It is not only a rectangle but also a square because right triangles ΔIHE,
ΔBJC, ΔJFE, ΔDIC are all congruent and their longer legs are its sides.</pre>and what is its area?<pre>Congruent right triangles ΔIHE, ΔBJC, ΔJFE, ΔDIC all have hypotenuses 2 and
shorter legs 1, so by the Pythagorean theorem, each side of square CIEJ is
√5.  Therefore the area of square CIEJ is 5.</pre>Is it possible to obtain a polygon with a larger area by slicing the cube
with a different plane? If so, show how to do it. If not, explain why it is
not possible.<pre>Yes, it is possible. CDEF and BGHA are rectangles with length 2√2 and width
2. So their areas are 4√2 which is approximately 5.656854249 > 5. 

{{{drawing(400,4800/13,-1,12,-1,11,

line(0,0,0,8),line(0,8,8,8),line(8,0,8,8),line(0,0,8,0),
line(3,10,11,10),line(11,2,11,10),line(3,2,11,2),line(3,10,3,2),
line(0,8,3,10),line(8,8,11,10),line(0,0,3,2),line(11,2,8,0),

green(line(3,10,1.5,1),line(1.5,1,8,0),line(9.5,9,8,0),line(3,10,9.5,9)),

red(line(8,8,3,10),line(8,8,8,0),line(3,2,8,0),line(3,10,3,2)),

blue(line(0,0,11,2),line(11,2,11,10),line(11,10,0,8),line(0,8,0,0)),

locate(0.05,8,A),locate(0,0,B),locate(8,0,C),locate(8.05,8,D),
locate(3.05,10,E),locate(3,2,F),locate(11,2,G),locate(11.05,10,H),
locate(9.5,9,I),locate(1.5,1,J), locate(3.9,0,2), locate(11.1,6,2),
locate(9.6,1,2)


)}}}

Edwin</pre>