Difference between revisions of "2023 AMC 12B Problems/Problem 25"
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<math>[Z] = \frac{1}{4\sin^{2}54}[ABCDE]</math> | <math>[Z] = \frac{1}{4\sin^{2}54}[ABCDE]</math> | ||
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<math> = \frac{2(1+\sqrt5)}{3+\sqrt5}</math> | <math> = \frac{2(1+\sqrt5)}{3+\sqrt5}</math> | ||
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<math> = \sqrt5-1</math> | <math> = \sqrt5-1</math> | ||
Revision as of 20:16, 15 November 2023
Problem
A regular pentagon with area 
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?
Solution 1
Let the original pentagon be 
centered at 
. The dashed lines represent the fold lines. WLOG, let's focus on vertex 
.
Since is folded onto , 
where 
is the intersection of 
and the creaseline between and . 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
Option 1: Knowledge
Remember that 
.
Option 2: Angle Identities
Let the inner pentagon be 
.
![$[Z] = \frac{1}{4\sin^{2}54}[ABCDE]$](http://latex.artofproblemsolving.com/8/c/6/8c6cef686dd33008c66f1d3dfdda458f5ad97bdb.png)
$\boxed{B 🐱}$ (Error compiling LaTeX. Unknown error_msg)
-Dissmo
