2012 AMC 12B Problems/Problem 19
Problem
A unit cube has vertices and
. Vertices
,
, and
are adjacent to
, and for
vertices
and
are opposite to each other. A regular octahedron has one vertex in each of the segments
,
,
,
,
, and
. What is the octahedron's side length?
Solution 1
Observe the diagram above. Each dot represents one of the six vertices of the regular octahedron. Three dots have been placed exactly x units from the corner of the unit cube. The other three dots have been placed exactly x units from the
corner of the unit cube. A red square has been drawn connecting four of the dots to provide perspective regarding the shape of the octahedron. Observe that the three dots that are near
are each (x)(
) from each other. The same is true for the three dots that are near
There is a unique
for which the rectangle drawn in red becomes a square. This will occur when the distance from
to
is
).
Using the distance formula we find the distance between the two points to be: =
. Equating this to
) and squaring both sides, we have the equation:
=
=
.
Since the length of each side is (x)(), we have a final result of
. Thus, Answer choice
is correct.
(If someone can draw a better diagram with the points labeled P1,P2, etc., I would appreciate it).
--Jm314 14:55, 26 February 2012 (EST)
Solution 2
Standard 3D geometry, no coordinates.
Let the tip of the octahedron on side be
and the opposite vertex be
. Our key is to examine the trapezoid
.
Let the side length of the octahedron be . Then
and
. Then, we have
. Finally, we want to find
, which is just double the height of half the octahedron. We can use Pythagorean Theorem to find that height as
. Now, we use the Pythagorean Theorem on the trapezoid. We get
~superagh
Solution 3
Let the length of ,
,
,
,
,
As ,
,
,
See Also
2012 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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