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

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pair A,B,C,D,E,F,W,X,Y,Z,M,N,O,P,Q,R;
 
pair A,B,C,D,E,F,W,X,Y,Z,M,N,O,P,Q,R;
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Revision as of 13:26, 18 February 2018

Problem

Let $ABCDEF$ be a regular hexagon with side length $1$. Denote by $X$, $Y$, 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 $\triangle XYZ$?

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

Solution 1

pair A,B,C,D,E,F,W,X,Y,Z,M,N,O,P,Q,R;
A=(0)


label("$A$",A,NW);
label("$B$",B,NE);
label("$C$",C,ESE);
label("$D$",D,SE);
label("$E$",E,SW);
label("$F$",F,WSW);
label("$X$", X, N);
label("$Y$", Y, ESE);
label("$Z$", Z, WSW);
label("$M$", M, NW);
label("$N$", N, NE);
label("$O$", O, SE);
label("$P$", P, NNW);
label("$Q$", Q, ESE);
label("$R$", R, SW);



 (Error compiling LaTeX. 9fc921d364cd52f73c58018d4d90e6373ea98113.asy: 9.1: syntax error
error: could not load module '9fc921d364cd52f73c58018d4d90e6373ea98113.asy')

The desired area (hexagon $MPNQOR$) consists of an equilateral triangle ($\triangle MNO$) and three right triangles ($\triangle MPN$, $\triangle NQO$, and $\triangle ORM$).

Notice that $\overline {AD}$ (not shown) and $\overline {BC}$ are parallel. $\overline {XY}$ divides transversals $\overline {AB}$ and $\overline {CD}$ into a $1:1$ ratio. Thus, it must also divide transversal $\overline {AC}$ and transversal $\overline {CO}$ into a $1:1$ ratio. By symmetry, the same applies for $\overline {CE}$ and $\overline {EA}$ as well as $\overline {EM}$ and $\overline {AN}$


In $\triangle ACE$, we see that $\frac{[MNO]}{[ACE]} = \frac{1}{4}$ and $\frac{[MPN]}{[ACE]} = \frac{1}{8}$. Our desired area becomes

\[(\frac{1}{4}+3 \cdot \frac{1}{8}) \cdot \frac{(\sqrt{3})^2 \cdot \sqrt{3}}{4} = \frac {15}{32}\sqrt{3} = \boxed {C}\]

~Llamabonobo

Solution 2

Now, if we look at the figure, we can see that the complement of the hexagon we are trying to find is composed of 3 isosceles trapezoids (AXFZ, XBCY, and ZYED), and 3 right triangles (With one vertice on each of X, Y, and Z). We know that one base of each trapezoid is just the side length of the hexagon which is 1, and the other base is 3/2 (It is halfway in between the side and the longest diagonal) with a height of $\sqrt{3}/4$ (by using the Pythagorean theorem and the fact that it is an isosceles trapezoid) to give each trapezoid having an area of $5\sqrt{3}/16$ for a total area of $15\sqrt{3}/16$ (Alternatively, we could have calculated the area of hexagon ABCDEF and subtracted the area of triangle XYZ, which, as we showed before, had a side length of 3/2). Now, we need to find the area of each of the small triangles, which, if we look at the triangle that has a vertice on X, is similar to the triangle with a base of YC = 1/2. Using similar triangles we calculate the base to be 1/4 and the height to be $\sqrt{3}/4$ giving us an area of $\sqrt{3}/32$ per triangle, and a total area of $3\sqrt{3}/32$. Adding the two areas together, we get $15\sqrt{3}/16+3\sqrt{3}/32=33*sqrt{3}/32$. Finding the total area, we get $6*1^2*\sqrt{3}/4=3\sqrt{3}/2$. Taking the complement, we get $3\sqrt{3}/2-33\sqrt{3}/32=15\sqrt{3}/32 = \boxed {C}$ --- Arpitr20

See Also

2018 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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
2018 AMC 12B (ProblemsAnswer KeyResources)
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 12 Problems and Solutions

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