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

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Since <math>\triangle BCD</math> and <math>\triangle BAD</math> are equilateral, <math>\ell_{BC}</math> contains <math>D</math>, <math>\ell_{BD}</math> contains <math>A</math> and <math>C</math>, and <math>\ell_{BA}</math> contains <math>D</math>. Then <math>\triangle BEF \cong \triangle BGF \cong \triangle BGH \cong \triangle BIH</math> with <math>BE = 1</math> and <math>EF = \frac{1}{\sqrt{3}}</math> so <math>[BEF] = \frac{1}{2}\cdot 1 \cdot \frac{\sqrt{3}}{3}</math>. Multiply this by 4 and it turns out that the pentagon has area <math>\boxed{(C)\frac{2\sqrt{3}}{3}}</math>.
 
Since <math>\triangle BCD</math> and <math>\triangle BAD</math> are equilateral, <math>\ell_{BC}</math> contains <math>D</math>, <math>\ell_{BD}</math> contains <math>A</math> and <math>C</math>, and <math>\ell_{BA}</math> contains <math>D</math>. Then <math>\triangle BEF \cong \triangle BGF \cong \triangle BGH \cong \triangle BIH</math> with <math>BE = 1</math> and <math>EF = \frac{1}{\sqrt{3}}</math> so <math>[BEF] = \frac{1}{2}\cdot 1 \cdot \frac{\sqrt{3}}{3}</math>. Multiply this by 4 and it turns out that the pentagon has area <math>\boxed{(C)\frac{2\sqrt{3}}{3}}</math>.
  
−
== Solution #2==
+
== Solution 2==
  
−
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{(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 <math>30-60-90</math> special right 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{(C)\frac{2\sqrt{3}}{3}}</math>.
 +
 
 +
== Solution 3==
 +
 
 +
We split rhombus <math>ABCD</math> into two equilateral triangles, <math>ABD</math> and <math>BCD</math>. In triangle <math>ABD</math>, the probability that a randomly selected point is closer to <math>B</math> than either other point is <math>\frac{1}{3}</math> (why?). Similarly, in triangle <math>BCD</math>, the same principle applies. Thus, the area of the region closer to <math>B</math> than <math>A</math>, <math>C</math>, or <math>D</math> is <math>\frac{1}{3} [ABD] + \frac{1}{3} [BCD]</math>. Since <math>ABD</math> and <math>BCD</math> are congruent, we have <math>\frac{1}{3} [ABD] + \frac{1}{3} [BCD] = \frac{2}{3} [ABD] = \frac{2}{3} \cdot \frac{s^2\sqrt{3}}{4} = \frac{2}{3} \cdot \frac{(2)^2\sqrt{3}}{4} = \boxed{\frac{2\sqrt3}{3} = C}</math>, and we are done.
 +
 
 +
== Solution 4 ==
 +
<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+B+D)/3, H=(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,S);label("$H$",H,E);
 +
label("$2$",(D--C),SW);
 +
</asy>
 +
Since <math>H</math> and <math>E</math> are halfway between <math>AB</math> and <math>CB</math>, respectively, we know that <math>\overline{BH}=\overline{BE}=1</math>. By symmetry, <math>\Delta BFG</math> is equilateral, so <math>\angle FBG=60^\circ\implies\angle EBF=\angle HBG=30^\circ</math> and therefore <math>\Delta EBF</math> and <math>\Delta HBG</math> are 30-60-90 right triangles.
 +
Thus, <math>[\Delta EBF]=[\Delta BFG]=\dfrac1{2\sqrt3}</math>.
 +
We know that <math>\overline{FB}=\overline{GB}=\dfrac2{\sqrt3}</math>, so therefore <math>[\Delta BFG]=\dfrac{\sqrt3}4\left(\dfrac2{\sqrt3}\right)^2=\dfrac1{\sqrt3}</math>.
 +
Summing these three regions, we get <math>\dfrac1{2\sqrt3}+\dfrac1{2\sqrt3}+\dfrac1{\sqrt3}=\boxed{\textbf{(C)}~\dfrac{2\sqrt3}3}</math>.
 +
~ Technodoggo, Asymptote diagram modified from Solution 1
 +
 
 +
== Solution 5 ==
 +
<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+B+D)/3, H=(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--F); draw(D--G); draw(F--C); draw(A--G); draw(F--B); draw(B--G);
 +
draw(B--D);
 +
 
 +
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,S);label("$H$",H,E);
 +
label("$2$",(D--C),SW);
 +
</asy>
 +
To keep it simple, break rhombus <math>ABCD</math> into two triangles, <math>ABD</math> and <math>BCD</math>. To see the area closest to the point <math>B</math>, notice that a third of each triangle, which contains all the points nearest to <math>B</math> in each triangle, is easily visualizable. Thus, a third of rhombus <math>ABCD</math> must be found.
 +
 
 +
We find the total area of rhombus <math>ABCD</math>, which we can again split into two congruent equilateral triangles with side length <math>2</math>. Using the formula of equilateral triangles and then multiplying by <math>\dfrac{1}{3}</math>:
 +
<cmath>\dfrac{\sqrt{3}}{4}\cdot 2^2 \cdot 2 \cdot \dfrac{1}{3} = 2\sqrt{3} \cdot \dfrac{1}{3} = \boxed{\textbf{(C)}~\dfrac{2\sqrt{3}}{3}}</cmath>
 +
-NSAoPS, diagram modified from Solution 1.
 +
 
 +
==Video Solution by TheBeautyofMath==
 +
https://youtu.be/gCmQlaiEG5A
 +
 
 +
~IceMatrix
  
 
== See Also==
 
== See Also==

Latest revision as of 20:05, 19 June 2024

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 $30-60-90$ special right 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 $2\sqrt {3}-\sqrt {3}-\frac{\sqrt{3}}{6}-\frac{\sqrt{3}}{6}=\boxed{(C)\frac{2\sqrt{3}}{3}}$.

Solution 3

We split rhombus $ABCD$ into two equilateral triangles, $ABD$ and $BCD$. In triangle $ABD$, the probability that a randomly selected point is closer to $B$ than either other point is $\frac{1}{3}$ (why?). Similarly, in triangle $BCD$, the same principle applies. Thus, the area of the region closer to $B$ than $A$, $C$, or $D$ is $\frac{1}{3} [ABD] + \frac{1}{3} [BCD]$. Since $ABD$ and $BCD$ are congruent, we have $\frac{1}{3} [ABD] + \frac{1}{3} [BCD] = \frac{2}{3} [ABD] = \frac{2}{3} \cdot \frac{s^2\sqrt{3}}{4} = \frac{2}{3} \cdot \frac{(2)^2\sqrt{3}}{4} = \boxed{\frac{2\sqrt3}{3} = C}$, and we are done.

Solution 4

[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+B+D)/3, H=(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,S);label("$H$",H,E); label("$2$",(D--C),SW); [/asy] Since $H$ and $E$ are halfway between $AB$ and $CB$, respectively, we know that $\overline{BH}=\overline{BE}=1$. By symmetry, $\Delta BFG$ is equilateral, so $\angle FBG=60^\circ\implies\angle EBF=\angle HBG=30^\circ$ and therefore $\Delta EBF$ and $\Delta HBG$ are 30-60-90 right triangles. Thus, $[\Delta EBF]=[\Delta BFG]=\dfrac1{2\sqrt3}$. We know that $\overline{FB}=\overline{GB}=\dfrac2{\sqrt3}$, so therefore $[\Delta BFG]=\dfrac{\sqrt3}4\left(\dfrac2{\sqrt3}\right)^2=\dfrac1{\sqrt3}$. Summing these three regions, we get $\dfrac1{2\sqrt3}+\dfrac1{2\sqrt3}+\dfrac1{\sqrt3}=\boxed{\textbf{(C)}~\dfrac{2\sqrt3}3}$. ~ Technodoggo, Asymptote diagram modified from Solution 1

Solution 5

[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+B+D)/3, H=(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--F); draw(D--G); draw(F--C); draw(A--G); draw(F--B); draw(B--G); draw(B--D); 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,S);label("$H$",H,E); label("$2$",(D--C),SW); [/asy] To keep it simple, break rhombus $ABCD$ into two triangles, $ABD$ and $BCD$. To see the area closest to the point $B$, notice that a third of each triangle, which contains all the points nearest to $B$ in each triangle, is easily visualizable. Thus, a third of rhombus $ABCD$ must be found.

We find the total area of rhombus $ABCD$, which we can again split into two congruent equilateral triangles with side length $2$. Using the formula of equilateral triangles and then multiplying by $\dfrac{1}{3}$: \[\dfrac{\sqrt{3}}{4}\cdot 2^2 \cdot 2 \cdot \dfrac{1}{3} = 2\sqrt{3} \cdot \dfrac{1}{3} = \boxed{\textbf{(C)}~\dfrac{2\sqrt{3}}{3}}\] -NSAoPS, diagram modified from Solution 1.

Video Solution by TheBeautyofMath

https://youtu.be/gCmQlaiEG5A

~IceMatrix

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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