Difference between revisions of "2023 AMC 12B Problems/Problem 25"

Revision as of 00:30, 16 November 2023

The following problem is from both the 2023 AMC 10B #25 and 2023 AMC 12B #25, so both problems redirect to this page.

Problem

A regular pentagon with area $\sqrt{5}+1$ is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon?

$\textbf{(A)}~4-\sqrt{5}\qquad\textbf{(B)}~\sqrt{5}-1\qquad\textbf{(C)}~8-3\sqrt{5}\qquad\textbf{(D)}~\frac{\sqrt{5}+1}{2}\qquad\textbf{(E)}~\frac{2+\sqrt{5}}{3}$

Solution 1

Let the original pentagon be $ABCDE$ centered at $O$ . The dashed lines represent the fold lines. WLOG, let's focus on vertex $A$ .

Since $A$ is folded onto $O$ , $AN = NO$ where $N$ is the intersection of $AO$ and the creaseline between $A$ and $O$ . Note that the inner pentagon is regular, and therefore similar to the original pentagon, due to symmetry.

Because of their similarity, the ratio of the inner pentagon's area to that of the outer pentagon can be represented by

$(\frac{ON}{OM})^{2} = (\frac{\frac{OA}{2}}{OA\sin (\angle OAE)})^{2} = \frac{1}{4\sin^{2}54}$

Option 1: Knowledge

Remember that $\sin54 = \frac{1+\sqrt5}{4}$ .

Option 2: Angle Identities

$\sin54 = \cos36$

$4\cos^{3}18-3\cos18 = 2\sin18\cos18$

$4(1-\sin^{2}18)-3-2\sin18=0$

$4\sin^{2}18+2\sin18-1=0$

$\sin18 = \frac{-1+\sqrt5}{4}$

$\sin54 = \cos36 = 1-2\sin^{2}18 = \frac{1+\sqrt5}{4}$

$\sin^{2}54 =\frac{3+\sqrt5}{8}$

Let the inner pentagon be $Z$ .

$[Z] = \frac{1}{4\sin^{2}54}[ABCDE]$

$= \frac{2(1+\sqrt5)}{3+\sqrt5}$

$= \sqrt5-1$

$\boxed{B}$

-Dissmo

Solution 2

$[asy] unitsize(5cm); // Define the vertices of the pentagons pair A, B, C, D, E; pair F, G, H, I, J; // Calculate the vertices of the larger pentagon A = dir(90); B = dir(90 - 72); C = dir(90 - 2*72); D = dir(90 - 3*72); E = dir(90 - 4*72); // Draw the larger pentagon draw(A--B--C--D--E--cycle); pair O = (A+B+C+D+E)/5; pair AA,OO; real gap = 0.02; AA = A+(0,0); OO = O+(0,0); draw(AA--OO, blue); pair OOO, OAO; OOO = O+(gap,0); OAO = (O+A)/2 + (gap,0); draw(OOO--OAO,green); dot(O); dot((O+A)/2); label("$r_b$", (O+A)*.7, E,blue); label("$a_s$", (O+A)*.2 +(0+0.18,0.05), E,green); label("$r_s$", O+(-0.175,0.2), E,pink); real scaleFactor = 1/1.618; // Adjust this value as needed // Rotate the smaller pentagon by 180 degrees F = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 + 180); G = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 - 72 + 180); H = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 - 2*72 + 180); I = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 - 3*72 + 180); J = (1 - scaleFactor) * (0,0) + scaleFactor * dir(90 - 4*72 + 180); // Draw the smaller pentagon draw(F--G--H--I--J--cycle,red); draw(arc(O,(H+I)*.5*.6,H*.6)); label("$36^\circ$",O+(+0.05,0.15),NW); draw(O--H,pink); [/asy]$

Let $r_b$ and $r_s$ be the circumradius of the big and small pentagon, respectively. Let $a_s$ be the apothem of the smaller pentagon and $A_s$ and $A_b$ be the areas of the smaller and larger pentagon, respectively.

From the diagram: $\begin{align*} \cos{36^\circ} &= \dfrac{a_s}{r_s} = \dfrac{\phi}{2} = \dfrac{\sqrt{5}+1}{4}\\ a_s &= \dfrac{r_b}{2}\\ A_s &= \left(\dfrac{r_s}{r_b}\right)^2A_b\\ &=\left(\dfrac{a_s}{\cos{36^\circ} r_b}\right)^2 \left(1+\sqrt{5}\right)\\ &=\left(\dfrac{r_b}{\dfrac{\phi}{2} r_b}\right)^2 \left(1+\sqrt{5}\right)\\ &=\left(\dfrac{1}{2 \dfrac{\phi}{2}}\right)^2 \left(1+\sqrt{5}\right)\\ &=\left(\dfrac{2}{\sqrt{5}+1}\right)^2 \left(1+\sqrt{5}\right)\\ &=\dfrac{4}{\sqrt{5}+1} \\ &=\dfrac{4\left(\sqrt{5}-1\right)}{\left(\sqrt{5}+1\right)\left(\sqrt{5}-1\right)} \\ &=\sqrt{5}-1 \end{align*}$ $\boxed{\textbf{(B) }\sqrt{5}-1}$ ~Technodoggo

Solution 3

Interestingly, we find that the pentagon we need is the one that is represented by the intersection of the diagonals. Through similar triangles and the golden ratio, we find that the side length ratio of the two pentagons is $\frac{\sqrt{5}-1}{2}$ Thus, the answer is $\sqrt{5}+1 \cdot (\frac{\sqrt{5}-1}{2})^2 = \sqrt{5}-1$ . $\boxed{\text{B}}$ ~andliu766

Supplement (Calculating sin54 from Scratch)

Construct $\triangle ABC$ with $\angle A = 36^{\circ}$ , $\angle B = \angle C = 72^{\circ}$ and $\triangle BCD$ with $\angle C = 36^{\circ}$ , $\angle B = \angle D = 72^{\circ}$ . WLOG, let $AC = 1$ , $BC = CD = AD = a$ , $BD = 1-a$ .

$\frac{AC}{BC} = \frac{BC}{BD}, \quad \frac{1}{a} = \frac{a}{1-a}, \quad 1-a=a^2, \quad a^2 + a - 1 = 0$

$a = \frac{ -1 + \sqrt{1^2 - 4(-1) } }{2} = \frac{ \sqrt{5} -1 }{2}$

$\cos 36^{\circ} = \cos \angle A = \frac{AE}{AC} = \frac{ 1-a }{2} + 1 = \frac{ a + 1 }{2} = \frac{ \frac{ \sqrt{5} -1 }{2} + 1 }{2} = \frac{ \sqrt{5} + 1 }{4}$

$\sin 54^{circ} = \cos 36^{\circ} = \frac{ \sqrt{5} + 1 }{4}$

~isabelchen

@@ Line 136: / Line 136: @@
 ==Supplement (Calculating sin54 from Scratch)==
-[[File:2023AMC12BP25.png|center|500px]]
+[[File:2023AMC12BP25.png|center|300px]]
-(Writing in progress......)
+Construct <math>\triangle ABC</math> with <math>\angle A = 36^{\circ}</math>, <math>\angle B = \angle C = 72^{\circ}</math> and <math>\triangle BCD</math> with <math>\angle C = 36^{\circ}</math>, <math>\angle B = \angle D = 72^{\circ}</math>. WLOG, let <math>AC = 1</math>, <math>BC = CD = AD = a</math>, <math>BD = 1-a</math>.
+<cmath>\frac{AC}{BC} = \frac{BC}{BD}, \quad \frac{1}{a} = \frac{a}{1-a}, \quad 1-a=a^2, \quad a^2 + a - 1 = 0</cmath>
+<cmath>a = \frac{ -1 + \sqrt{1^2  - 4(-1) } }{2} = \frac{ \sqrt{5} -1 }{2}</cmath>
+<cmath>\cos 36^{\circ} = \cos \angle A = \frac{AE}{AC} = \frac{ 1-a }{2} + 1 = \frac{ a + 1 }{2} = \frac{ \frac{ \sqrt{5} -1 }{2} + 1 }{2} = \frac{ \sqrt{5} + 1 }{4}</cmath>
+<cmath>\sin 54^{circ} = \cos 36^{\circ} = \frac{ \sqrt{5} + 1 }{4}</cmath>
 ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

2023 AMC 10B (Problems • Answer Key • Resources)
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