Difference between revisions of "2024 AMC 12B Problems/Problem 23"

Revision as of 15:32, 14 November 2024

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

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

~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

@@ Line 28: / Line 28: @@
 [[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
 ==See also==
 {{AMC12 box|year=2024|ab=B|num-b=22|num-a=24}}
 {{MAA Notice}}

2024 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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