1995 AIME Problems/Problem 12
Pyramid has square base congruent edges and and Let be the measure of the dihedral angle formed by faces and Given that where and are integers, find
Solution 1 (trigonometry)
The angle is the angle formed by two perpendiculars drawn to , one on the plane determined by and the other by . Let the perpendiculars from and to meet at Without loss of generality, let It follows that is a right triangle, so and Therefore,
From the Law of Cosines, so
Solution 2 (analytical/vectors)
Without loss of generality, place the pyramid in a 3-dimensional coordinate system such that and where is unknown.
We first find Note that
Since and this simplifies to
Now let's find Let and be normal vectors to the planes containing faces and respectively. From the definition of the dot product as , we will be able to solve for A cross product yields (alternatively, it is simple to find the equation of the planes and , and then to find their normal vectors)
Hence, taking the dot product of and yields
Flipping the signs (we found the cosine of the supplement angle) yields so the answer is .
Solution 3 (bashy trig)
Similar to Solution 1, is the dihedral angle we want. WLOG, we will let meaning .
Because are isosceles, .
Thus by the half-angle identity,
Now looking at triangle we drop the perpendicular from to , and call the foot . Then By Pythagoreas,
We have that
Because and can be negative integers, our answer is
Notice that as well.
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