Difference between revisions of "2018 AMC 10B Problems/Problem 24"

Revision as of 16:29, 16 February 2018

Problem

Let $ABCDEF$ be a regular hexagon with side length $1$ . Denote $X$ , $Y$ , and $Z$ the midpoints of sides $\overline {AB}$ , $\overline{CD}$ , and $\overline{EF}$ , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$ ?

$\textbf{(A)} \frac {3}{8}\sqrt{3} \qquad \textbf{(B)} \frac {7}{16}\sqrt{3} \qquad \textbf{(C)} \frac {15}{32}\sqrt{3} \qquad \textbf{(D)} \frac {1}{2}\sqrt{3} \qquad \textbf{(E)} \frac {9}{16}\sqrt{3} \qquad$

Answer: $\frac {15}{32}\sqrt{3}$

Solution

$[asy] pair A,B,C,D,E,F,W,X,Y,Z; A=(0,0); B=(1,0); C=(3/2,sqrt(3)/2); D=(1,sqrt(3)); E=(0,sqrt(3)); F=(-1/2,sqrt(3)/2); W=(4/3,2sqrt(3)/3); X=(4/3,sqrt(3)/3); Y=(-1/3,sqrt(3)/3); Z=(-1/3,2sqrt(3)/3); draw(A--B--C--D--E--F--cycle); draw(W--Z); draw(X--Y); draw(F--C--B--E--D--A); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,ESE); label("$D$",D,NE); label("$E$",E,NW); label("$F$",F,WSW); label("$W$",W,ENE); label("$X$",X,ESE); label("$Y$",Y,WSW); label("$Z$",Z,WNW); [/asy]$

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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All AMC 10 Problems and Solutions

2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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All AMC 12 Problems and Solutions

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Difference between revisions of "2018 AMC 10B Problems/Problem 24"

Revision as of 16:29, 16 February 2018

Problem

Solution

See Also

@@ Line 11: / Line 11: @@
 <asy>
+pair A,B,C,D,E,F,W,X,Y,Z;
+A=(0,0);
+B=(1,0);
+C=(3/2,sqrt(3)/2);
+D=(1,sqrt(3));
+E=(0,sqrt(3));
+F=(-1/2,sqrt(3)/2);
+W=(4/3,2sqrt(3)/3);
+X=(4/3,sqrt(3)/3);
+Y=(-1/3,sqrt(3)/3);
+Z=(-1/3,2sqrt(3)/3);
+draw(A--B--C--D--E--F--cycle);
+draw(W--Z);
+draw(X--Y);
+draw(F--C--B--E--D--A);
+label("$A$",A,SW);
+label("$B$",B,SE);
+label("$C$",C,ESE);
+label("$D$",D,NE);
+label("$E$",E,NW);
+label("$F$",F,WSW);
+label("$W$",W,ENE);
+label("$X$",X,ESE);
+label("$Y$",Y,WSW);
+label("$Z$",Z,WNW);
 </asy>