2016 AIME I Problems/Problem 4
Contents
Problem
A right prism with height has bases that are regular hexagons with sides of length . A vertex of the prism and its three adjacent vertices are the vertices of a triangular pyramid. The dihedral angle (the angle between the two planes) formed by the face of the pyramid that lies in a base of the prism and the face of the pyramid that does not contain measures degrees. Find .
Diagram
~gundraja
Solution 1
Let and be the vertices adjacent to on the same base as , and let be the last vertex of the triangular pyramid. Then . Let be the foot of the altitude from to . Then since is a triangle, . Since the dihedral angle between and is , is a triangle and . Thus .
~gundraja
Solution 2
Let and be the vertices adjacent to on the same base as , and let be the last vertex of the triangular pyramid. Notice that we can already find some lengths. We have (given) and by the Pythagorean Theorem. Let be the midpoint of . Then, we have () triangles and by the Pythagorean Theorem. Applying the Law of Cosines, since , we get as desired.
-A1001
See also
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.