2024 AIME I Problems/Problem 14
Solution 1
Notice that , , and , let , , , and . Then the plane has a normal
Hence, the distance from to plane , or the height of the tetrahedron, is
Each side of the tetrahedron has the same area due to congruency by "S-S-S", and we call it . Then by the volume formula for cones,
Solution by Quantum-Phantom
Solution 2
Inscribe tetrahedron in an rectangular prism as shown above.
By the Pythagorean theorem, we note
Solving yields and
Since each face of the tetrahedron is congruent, we know the point we seek is the center of the circumsphere of We know all rectangular prisms can be inscribed in a circumsphere, therefore the circumsphere of the rectangular prism is also the circumsphere of
We know that the distance from all faces must be the same, so we only need to find the distance from the center to plane .
Let and We obtain that the plane of can be marked as or and the center of the prism is
Using the Point-to-Plane distance formula, our distance is
Our answer is
- spectraldragon8
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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