2011 AIME I Problems/Problem 13
A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled . The three vertices adjacent to vertex are at heights 10, 11, and 12 above the plane. The distance from vertex to the plane can be expressed as , where , , and are positive integers, and . Find .
Set the cube at the origin with the three vertices along the axes and the plane equal to , where . The distance from a point to a plane with equation is so the (directed) distance from any point to the plane is . So, by looking at the three vertices, we have , and by rearranging and summing,
Solving the equation is easier if we substitute , to get , or . The distance from the origin to the plane is simply , which is equal to , so .
Set the cube at the origin and the adjacent vertices as (10, 0, 0), (0, 10, 0) and (0, 0, 10). Then consider the plane ax + by + cz = 0. Because A has distance 0 to it (and distance d to the original, parallel plane), the distance from the other vertices to the plane is 10-d, 11-d, and 12-d respectively. The distance formula gives and Squaring each equation and then adding yields , and we can proceed as in the first solution.
Let the vertices with distance be , respectively. An equilateral triangle is formed with side length . We care only about the coordinate: . It is well known that the centroid of a triangle is the average of the coordinates of its three vertices, so . Designate the midpoint of as . Notice that median is parallel to the plane because the and vertex have the same coordinate, , and the median contains and the . We seek the angle of the line: through the centroid perpendicular to the plane formed by , with the plane under the cube. Since the median is parallel to the plane, this orthogonal line is also perpendicular to . Since makes a right triangle, the orthogonal line makes the same right triangle rotated . Therefore, .
It is also known that the centroid of is a third of the way between vertex and , the vertex farthest from the plane. Since is a diagonal of the cube, . So the distance from the to is . So, the from to the centroid is .
Thus the distance from to the plane is , and .
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