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Difference between revisions of "2024 AMC 12B Problems/Problem 23"
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From symmetry, we know that <math>\overline{AV} = \overline{DV}</math>, therefore <math>\triangle{AVD}</math> is a 45-45-90 triangle. Denote <math>\overline{AV}</math> as <math>x</math> so that <math>\overline{AD} = x\sqrt{2}</math>. Doing some geometry on the isosceles trapezoid <math>ABCD</math> (we know this from the fact that it is a regular octagon) reveals that <math>\overline{AD}=1+2(\sqrt{2}/2)=1+\sqrt{2}</math> and <math>\overline{AV}=(\overline{AD})/\sqrt{2}=(\sqrt{2}+2)/2</math>.  
 
From symmetry, we know that <math>\overline{AV} = \overline{DV}</math>, therefore <math>\triangle{AVD}</math> is a 45-45-90 triangle. Denote <math>\overline{AV}</math> as <math>x</math> so that <math>\overline{AD} = x\sqrt{2}</math>. Doing some geometry on the isosceles trapezoid <math>ABCD</math> (we know this from the fact that it is a regular octagon) reveals that <math>\overline{AD}=1+2(\sqrt{2}/2)=1+\sqrt{2}</math> and <math>\overline{AV}=(\overline{AD})/\sqrt{2}=(\sqrt{2}+2)/2</math>.  
  
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To find the length <math>\overline{IA}</math>, we cut the octagon into 8 triangles, each witha smallest angle of 45 degrees. Using the law of cosines on <math>\triangle{AIB}</math> we find that <math>{\overline{IA}}^2=(2+\sqrt{2})/2</math>.  
+
To find the length <math>\overline{IA}</math>, we cut the octagon into 8 triangles, each with a smallest angle of 45 degrees. Using the law of cosines on <math>\triangle{AIB}</math> we find that <math>{\overline{IA}}^2=(2+\sqrt{2})/2</math>.  
  
 
Finally, using the pythagorean theorem, we can find that <math>{\overline{IV}}^2={\overline{AV}}^2-{\overline{IA}}^2= {((\sqrt{2}+2)/2)}^2 - (2+\sqrt{2})/2 = \boxed{(1+\sqrt{2})/2}</math> which is answer choice <math>\boxed{B}</math>.
 
Finally, using the pythagorean theorem, we can find that <math>{\overline{IV}}^2={\overline{AV}}^2-{\overline{IA}}^2= {((\sqrt{2}+2)/2)}^2 - (2+\sqrt{2})/2 = \boxed{(1+\sqrt{2})/2}</math> which is answer choice <math>\boxed{B}</math>.
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 +
～username2333
 +
~hashbrown2009
  
 
==Solution 2 (Less computation)==
 
==Solution 2 (Less computation)==
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<cmath>=\frac{1+\sqrt{2}}{2}</cmath>  
 
<cmath>=\frac{1+\sqrt{2}}{2}</cmath>  
 
Hence, the answer is <math>\boxed{B}</math>.
 
Hence, the answer is <math>\boxed{B}</math>.
 +
 +
~tsun26
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==Solution 3==
 +
[[Image: 2024_AMC_12B_P23.jpeg|thumb|center|600px|]]
 +
~Kathan
 +
 +
==Solution 4 (Vectors)==
 +
 +
Consider the vectors <math>\vec{AV}</math> and <math>\vec{DV}</math>.
 +
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
 +
<cmath>\vec{AV} = \left\langle \frac{1}{2},  \frac{1+\sqrt{2}}{2}, h \right\rangle</cmath>
 +
<cmath>\vec{DV} = \left\langle \frac{1}{2},  \frac{-1-\sqrt{2}}{2}, h \right\rangle</cmath>
 +
Now, we must have <math>\vec{AV} \cdot \vec{DV} = 0</math> if the vectors are perpendicular to each other, so
 +
<cmath>\left\langle \frac{1}{2},  \frac{-1-\sqrt{2}}{2}, h \right\rangle \cdot \left\langle \frac{1}{2},  \frac{1+\sqrt{2}}{2}, h \right\rangle = \frac{1}{4} - \frac{3 + 2\sqrt{2}}{4} + h^2 = 0</cmath>
 +
<cmath>h^2=\frac{2+2\sqrt{2}}{4}=\frac{1+\sqrt{2}}{2}</cmath>
 +
 +
Yielding answer choice <math>\boxed{B}</math>.
 +
 +
~tkl
 +
 +
==Only B and D looks normal , guess one using掐头去尾
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 +
==Video Solution 1 by SpreadTheMathLove==
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https://www.youtube.com/watch?v=G-PzTyKqqV4
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 +
==See also==
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{{AMC12 box|year=2024|ab=B|num-b=22|num-a=24}}
 +
{{MAA Notice}}

Latest revision as of 17:50, 16 March 2025

Problem

A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?

$\textbf{(A) }1 \qquad \textbf{(B) }\frac{1+\sqrt2}{2} \qquad \textbf{(C) }\sqrt2 \qquad \textbf{(D) }\frac32 \qquad \textbf{(E) }\frac{2+\sqrt2}{3} \qquad$


Solution 1

To find the height of the pyramid, we need the length from the center of the octagon (denote as $I$) to its vertices and the length of AV.

From symmetry, we know that $\overline{AV} = \overline{DV}$, therefore $\triangle{AVD}$ is a 45-45-90 triangle. Denote $\overline{AV}$ as $x$ so that $\overline{AD} = x\sqrt{2}$. Doing some geometry on the isosceles trapezoid $ABCD$ (we know this from the fact that it is a regular octagon) reveals that $\overline{AD}=1+2(\sqrt{2}/2)=1+\sqrt{2}$ and $\overline{AV}=(\overline{AD})/\sqrt{2}=(\sqrt{2}+2)/2$.

To find the length $\overline{IA}$, we cut the octagon into 8 triangles, each with a smallest angle of 45 degrees. Using the law of cosines on $\triangle{AIB}$ we find that ${\overline{IA}}^2=(2+\sqrt{2})/2$.

Finally, using the pythagorean theorem, we can find that ${\overline{IV}}^2={\overline{AV}}^2-{\overline{IA}}^2= {((\sqrt{2}+2)/2)}^2 - (2+\sqrt{2})/2 = \boxed{(1+\sqrt{2})/2}$ which is answer choice $\boxed{B}$.

～username2333 ~hashbrown2009

Solution 2 (Less computation)

Let $O$ be the center of the regular octagon. Connect $AD$, and let $I$ be the midpoint of line segment $AD$. It is easy to see that $VI=\frac{1}{2} AD=\frac{1+\sqrt{2}}{2}$ and $OI=\frac{1}{2}AH=\frac{1}{4}$. Hence, \[VO^2=VI^2-OI^2\] \[=\left(\frac{1+\sqrt{2}}{2}\right)^2-\frac{1}{4}\] \[=\frac{1+\sqrt{2}}{2}\] Hence, the answer is $\boxed{B}$.

~tsun26

Solution 3

2024 AMC 12B P23.jpeg

~Kathan

Solution 4 (Vectors)

Consider the vectors $\vec{AV}$ and $\vec{DV}$. 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 \[\vec{AV} = \left\langle \frac{1}{2}, \frac{1+\sqrt{2}}{2}, h \right\rangle\] \[\vec{DV} = \left\langle \frac{1}{2}, \frac{-1-\sqrt{2}}{2}, h \right\rangle\] Now, we must have $\vec{AV} \cdot \vec{DV} = 0$ if the vectors are perpendicular to each other, so \[\left\langle \frac{1}{2}, \frac{-1-\sqrt{2}}{2}, h \right\rangle \cdot \left\langle \frac{1}{2}, \frac{1+\sqrt{2}}{2}, h \right\rangle = \frac{1}{4} - \frac{3 + 2\sqrt{2}}{4} + h^2 = 0\] \[h^2=\frac{2+2\sqrt{2}}{4}=\frac{1+\sqrt{2}}{2}\]

Yielding answer choice $\boxed{B}$.

~tkl

==Only B and D looks normal , guess one using掐头去尾

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=G-PzTyKqqV4

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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