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2021 AMC 10B Problems/Problem 21: Difference between revisions

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hi im problem 211
==Problem==
[url=https://aops.com/community/p20334805][size=150][b]Problem 21[/b][/size][/url]
A square piece of paper has side length <math>1</math> and vertices <math>A,B,C,</math> and <math>D</math> in that order. As shown in the figure, the paper is folded so that vertex <math>C</math> meets edge <math>\overline{AD}</math> at point <math>C'</math>, and edge <math>\overline{AB}</math> at point <math>E</math>. Suppose that <math>C'D = \frac{1}{3}</math>. What is the perimeter of triangle <math>\bigtriangleup AEC' ?</math>
 
<math>\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\sqrt{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}</math>
<asy>
pair A=(0,1);
pair CC=(0.666666666666,1);
pair D=(1,1);
pair F=(1,0.62);
pair C=(1,0);
pair B=(0,0);
pair G=(0,0.25);
pair H=(-0.13,0.41);
pair E=(0,0.5);
dot(A^^CC^^D^^C^^B^^E);
draw(E--A--D--F);
draw(G--B--C--F, dashed);
fill(E--CC--F--G--H--E--CC--cycle, gray);
draw(E--CC--F--G--H--E--CC);
label("A",A,NW);
label("B",B,SW);
label("C",C,SE);
label("D",D,NE);
label("E",E,NW);
label("C",CC,N);
</asy>
==Solution==
A

Revision as of 18:28, 11 February 2021

Problem

[url=https://aops.com/community/p20334805][size=150][b]Problem 21[/b][/size][/url] A square piece of paper has side length $1$ and vertices $A,B,C,$ and $D$ in that order. As shown in the figure, the paper is folded so that vertex $C$ meets edge $\overline{AD}$ at point $C'$, and edge $\overline{AB}$ at point $E$. Suppose that $C'D = \frac{1}{3}$. What is the perimeter of triangle $\bigtriangleup AEC' ?$

$\textbf{(A)} ~2 \qquad\textbf{(B)} ~1+\frac{2}{3}\sqrt{3} \qquad\textbf{(C)} ~\sqrt{13}{6} \qquad\textbf{(D)} ~1 + \frac{3}{4}\sqrt{3} \qquad\textbf{(E)} ~\frac{7}{3}$ [asy] pair A=(0,1); pair CC=(0.666666666666,1); pair D=(1,1); pair F=(1,0.62); pair C=(1,0); pair B=(0,0); pair G=(0,0.25); pair H=(-0.13,0.41); pair E=(0,0.5); dot(A^^CC^^D^^C^^B^^E); draw(E--A--D--F); draw(G--B--C--F, dashed); fill(E--CC--F--G--H--E--CC--cycle, gray); draw(E--CC--F--G--H--E--CC); label("A",A,NW); label("B",B,SW); label("C",C,SE); label("D",D,NE); label("E",E,NW); label("C",CC,N); [/asy]

Solution

A

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