2006 AIME I Problems/Problem 14
A tripod has three legs each of length feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then can be written in the form where and are positive integers and is not divisible by the square of any prime. Find (The notation denotes the greatest integer that is less than or equal to )
We will use to denote volume (four letters), area (three letters) or length (two letters).
Let be the top of the tripod, are end points of three legs. Let be the point on such that and . Let be the center of the base equilateral triangle . Let be the midpoint of segment . Let be the distance from to the triangle ( is what we want to find).
We have the volume ratio .
We also have the area ratio .
The triangle is a right triangle so and .
Applying Law of Cosines to the triangle with , and , we find:
Putting it all together, we find .
We note that . From this we can derive that the side length of the equilateral is . We now use 3D coordinate geometry. We know three points of plane hence we can write out the equation for the plane. Plane can be expressed as
Applying the distance between a point and a plane formula.
Solution by SimonSun
Solution 3 (Cosine Law & Pythagorean Bash)
Diagram borrowed from Solution 1
Apply Pythagorean Theorem on yields Since is equilateral, we have and Apply Pythagorean Theorem on yields Apply Law of Cosines on we have Apply Law of Cosines on using the fact that we have Apply Pythagorean Theorem on yields Let the perpendicular from hits at . Let and . Apply Pythagorean Theorem on and we have Cancelling out the term and solving gets .
Finally, by Pythagorean Theorem, so
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