1995 AIME Problems/Problem 12
Problem
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
Contents
[hide]Solution
Solution 1 (trigonometry)
![[asy] import three; triple A = (1,0,0), B=(0,0,0), C=(0,1,0), D=(1,1,0), O=(1,1,(1+2^.5)^.5)/2^.5, P=O*(18^.5-2)/5; /* , P = foot(A, O, B) */ draw(A--B--C--D--A--O--B--O--C--O--D); D(A--P--C); [/asy]](http://latex.artofproblemsolving.com/8/2/d/82d0ff9d0b72ad0d7e154c81616a81d0369627f1.png)
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
and
Therefore,
From the Law of Cosines, so
Thus .
Solution 2 (analytic/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. It follows that letting
will allow us to solve for A cross product yields
Similarly,
Hence, taking the dot product of and
yields
Simplifying,
Flipping the signs (we found the cosine of the supplement angle) yields so the answer is
.
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |