2012 AMC 12B Problems/Problem 19
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?
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 ) 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 ), 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)
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
Let the length of ,
As , , ,
|2012 AMC 12B (Problems • Answer Key • Resources)|
|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|