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Difference between revisions of "2011 AMC 10B Problems/Problem 20"

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== Solution #2==
 
== Solution #2==
  
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We follow the steps shown above until we draw pentagon <math>BIHFE</math>. We know that rhombus <math>ABCD</math> can be divided into equilateral triangles <math>\triangle ABD</math> and <math>\triangle CBD</math>. Using the 30-60-90 triangle rules, we find the height of the equilateral triangles (and the height of the rhombus) to be <math>\sqrt{3}</math>. Therefore, the area of <math>ABCD</math> is <math>2\sqrt{3}</math>. We now have to take off the areas <math>\triangle CDA</math>, <math>\triangle CEF</math>, and <math>\triangle AIH</math> to get the desired shape. <math>\triangle CDA</math> is just half of <math>ABCD</math> <math>(\sqrt {3})</math> and <math>\triangle AIH</math> and <math>\triangle CEF</math> are each <math>\frac{\sqrt {3}}{6}</math>, for a total area of <math>2\sqrt {3}-\sqrt {3}-\frac{\sqrt{3}}{6}-\frac{\sqrt{3}}{6}=\boxed\boxed{(C)\frac{2\sqrt{3}}{3}}</math>.
+
We follow the steps shown above until we draw pentagon <math>BIHFE</math>. We know that rhombus <math>ABCD</math> can be divided into equilateral triangles <math>\triangle ABD</math> and <math>\triangle CBD</math>. Using the 30-60-90 triangle rules, we find the height of the equilateral triangles (and the height of the rhombus) to be <math>\sqrt{3}</math>. Therefore, the area of <math>ABCD</math> is <math>2\sqrt{3}</math>. We now have to take off the areas <math>\triangle CDA</math>, <math>\triangle CEF</math>, and <math>\triangle AIH</math> to get the desired shape. <math>\triangle CDA</math> is just half of <math>ABCD</math> <math>(\sqrt {3})</math> and <math>\triangle AIH</math> and <math>\triangle CEF</math> are each <math>\frac{\sqrt {3}}{6}</math>, for a total area of <math>2\sqrt {3}-\sqrt {3}-\frac{\sqrt{3}}{6}-\frac{\sqrt{3}}{6}=\boxed{\boxed{(C)\frac{2\sqrt{3}}{3}}}</math>.
  
 
== See Also==
 
== See Also==

Revision as of 20:46, 16 March 2018

The following problem is from both the 2011 AMC 12B #16 and 2011 AMC 10B #20, so both problems redirect to this page.

Problem

Rhombus $ABCD$ has side length $2$ and $\angle B = 120$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?

$\textbf{(A)}\ \frac{\sqrt{3}}{3} \qquad\textbf{(B)}\ \frac{\sqrt{3}}{2} \qquad\textbf{(C)}\ \frac{2\sqrt{3}}{3} \qquad\textbf{(D)}\ 1 + \frac{\sqrt{3}}{3} \qquad\textbf{(E)}\ 2$

Solution

Suppose that $P$ is a point in the rhombus $ABCD$ and let $\ell_{BC}$ be the perpendicular bisector of $\overline{BC}$. Then $PB < PC$ if and only if $P$ is on the same side of $\ell_{BC}$ as $B$. The line $\ell_{BC}$ divides the plane into two half-planes; let $S_{BC}$ be the half-plane containing $B$. Let us define similarly $\ell_{BD},S_{BD}$ and $\ell_{BA},S_{BA}$. Then $R$ is equal to $ABCD \cap S_{BC} \cap S_{BD} \cap S_{BA}$. The region turns out to be an irregular pentagon. We can make it easier to find the area of this region by dividing it into four triangles:

[asy] unitsize(8mm); defaultpen(linewidth(0.8pt)+fontsize(10pt)); dotfactor=4; pair A=(4,0), B=(2,2sqrt(3)), C=(-2,2sqrt(3)), D=(0,0), E=(B+C)/2, F=(B+C+D)/3, G=(A+C)/2, H=(A+B+D)/3, I=(A+B)/2; fill((0,2sqrt(3))--B--(3,sqrt(3))--(2,(2sqrt(3))/3)--(0,(4sqrt(3))/3)--cycle,gray); draw(A--B--C--D--cycle); draw(D--(0,2sqrt(3))); draw(D--(3,sqrt(3))); draw(A--C); draw(F--B--H); draw(B--G); label("$A$",A,SE);label("$B$",B,NE);label("$C$",C,NW);label("$D$",D,SW); label("$E$",E,N);label("$F$",F,SW);label("$G$",G,SW);label("$H$",H,S);label("$I$",I,NE); label("$2$",(D--C),SW); [/asy] Since $\triangle BCD$ and $\triangle BAD$ are equilateral, $\ell_{BC}$ contains $D$, $\ell_{BD}$ contains $A$ and $C$, and $\ell_{BA}$ contains $D$. Then $\triangle BEF \cong \triangle BGF \cong \triangle BGH \cong \triangle BIH$ with $BE = 1$ and $EF = \frac{1}{\sqrt{3}}$ so $[BEF] = \frac{1}{2}\cdot 1 \cdot \frac{\sqrt{3}}{3}$. Multiply this by 4 and it turns out that the pentagon has area $\boxed{(C)\frac{2\sqrt{3}}{3}}$.

Solution #2

We follow the steps shown above until we draw pentagon $BIHFE$. We know that rhombus $ABCD$ can be divided into equilateral triangles $\triangle ABD$ and $\triangle CBD$. Using the 30-60-90 triangle rules, we find the height of the equilateral triangles (and the height of the rhombus) to be $\sqrt{3}$. Therefore, the area of $ABCD$ is $2\sqrt{3}$. We now have to take off the areas $\triangle CDA$, $\triangle CEF$, and $\triangle AIH$ to get the desired shape. $\triangle CDA$ is just half of $ABCD$ $(\sqrt {3})$ and $\triangle AIH$ and $\triangle CEF$ are each $\frac{\sqrt {3}}{6}$, for a total area of $2\sqrt {3}-\sqrt {3}-\frac{\sqrt{3}}{6}-\frac{\sqrt{3}}{6}=\boxed{\boxed{(C)\frac{2\sqrt{3}}{3}}}$.

See Also

2011 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2011 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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