2024 AMC 12B Problems/Problem 23
Contents
[hide]Problem
A right pyramid has regular octagon with side length as its base and apex Segments and are perpendicular. What is the square of the height of the pyramid?
Solution 1
To find the height of the pyramid, we need the length from the center of the octagon (denote as ) to its vertices and the length of AV.
From symmetry, we know that , therefore is a 45-45-90 triangle. Denote as so that . Doing some geometry on the isosceles trapezoid (we know this from the fact that it is a regular octagon) reveals that and .
To find the length , we cut the octagon into 8 triangles, each with a smallest angle of 45 degrees. Using the law of cosines on we find that .
Finally, using the pythagorean theorem, we can find that which is answer choice .
~username2333
Solution 2 (Less computation)
Let be the center of the regular octagon. Connect , and let be the midpoint of line segment . It is easy to see that and . Hence, Hence, the answer is .
~tsun26
Solution 3
~Kathan
Solution 4 (Vectors)
Consider the vectors and . If we use a coordinate plane where one of the axes is parallel to one of the sides of the octagon, we can calculate each of the vectors to be Now, we must have if the vectors are perpendicular to each other, so
Yielding answer choice .
~tkl
See also
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 22 |
Followed by Problem 24 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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