Difference between revisions of "2003 AIME II Problems/Problem 14"

Revision as of 15:02, 19 February 2020

Problem

Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB = 120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m + n.$

Solution 1

The y-coordinate of $F$ must be $4$ . All other cases yield non-convex and/or degenerate hexagons, which violate the problem statement.

Letting $F = (f,4)$ , and knowing that $\angle FAB = 120^\circ$ , we can use rewrite $F$ using complex numbers: $f + 4 i = (b + 2 i)\left(e^{i(2 \pi / 3)}\right) = (b + 2 i)\left(-1/2 + \frac{\sqrt{3}}{2} i\right) = -\frac{b}{2}-\sqrt{3}+\left(\frac{b\sqrt{3}}{2}-1\right)i$ . We solve for $b$ and $f$ and find that $F = \left(-\frac{8}{\sqrt{3}}, 4\right)$ and that $B = \left(\frac{10}{\sqrt{3}}, 2\right)$ .

The area of the hexagon can then be found as the sum of the areas of two congruent triangles ( $EFA$ and $BCD$ , with height $8$ and base $\frac{8}{\sqrt{3}}$ ) and a parallelogram ( $ABDE$ , with height $8$ and base $\frac{10}{\sqrt{3}}$ ).

$A = 2 \times \frac{1}{2} \times 8 \times \frac{8}{\sqrt{3}} + 8 \times \frac{10}{\sqrt{3}} = \frac{144}{\sqrt{3}} = 48\sqrt{3}$ .

Thus, $m+n = \boxed{051}$ .

Solution 2

$[asy] size(200); draw((0,0)--(10/sqrt(3),2)--(18/sqrt(3),6)--(10/sqrt(3),10)--(0,8)--(-8/sqrt(3),4)--cycle); dot((0,0));dot((10/sqrt(3),2));dot((18/sqrt(3),6));dot((10/sqrt(3),10));dot((0,8));dot((-8/sqrt(3),4)); label("$A (0,0)$",(0,0),S);label("$B (b,2)$",(10/sqrt(3),2),SE);label("$C$",(18/sqrt(3),6),E);label("$D$",(10/sqrt(3),10),N);label("$E$",(0,8),NW);label("$F$",(-8/sqrt(3),4),W); [/asy]$ From this image, we can see that the y-coordinate of F is 4, and from this, we can gather that the coordinates of E, D, and C, respectively, are 8, 10, and 6.

$[asy] size(200); draw((0,0)--(10/sqrt(3),2)--(18/sqrt(3),6)--(10/sqrt(3),10)--(0,8)--(-8/sqrt(3),4)--cycle); dot((0,0));dot((10/sqrt(3),2));dot((18/sqrt(3),6));dot((10/sqrt(3),10));dot((0,8));dot((-8/sqrt(3),4)); label("$A (0,0)$",(0,0),SE);label("$B (b,2)$",(10/sqrt(3),2),SE);label("$C$",(18/sqrt(3),6),E);label("$D$",(10/sqrt(3),10),N);label("$E$",(0,8),NW);label("$F$",(-8/sqrt(3),4),W); xaxis("$x$");yaxis("$y$"); pair b=foot((10/sqrt(3),2),(0,0),(10,0)); pair f=foot((-8/sqrt(3),4),(0,0),(-10,0)); draw(b--(10/sqrt(3),2),dotted); draw(f--(-8/sqrt(3),4),dotted); label("$\theta$",(0,0),7*dir((0,0)--(10/sqrt(3),2)+(4*sqrt(21)/3,0))); [/asy]$

Let the angle between the $x$ -axis and segment $AB$ be $\theta$ , as shown above. Thus, as $\angle FAB=120^\circ$ , the angle between the $x$ -axis and segment $AF$ is $60-\theta$ , so $\sin{(60-\theta)}=2\sin{\theta}$ . Expanding, we have

$\sin{60}\cos{\theta}-\cos{60}\sin{\theta}=\frac{\sqrt{3}\cos{\theta}}{2}-\frac{\sin{\theta}}{2}=2\sin{\theta}$

Isolating $\sin{\theta}$ we see that $\frac{\sqrt{3}\cos{\theta}}{2}=\frac{5\sin{\theta}}{2}$ , or $\cos{\theta}=\frac{5}{\sqrt{3}}\sin{\theta}$ . Using the fact that $\sin^2{\theta}+\cos^2{\theta}=1$ , we have $\frac{28}{3}\sin^2{\theta}=1$ , or $\sin{\theta}=\sqrt{\frac{3}{28}}$ . Letting the side length of the hexagon be $y$ , we have $\frac{2}{y}=\sqrt{\frac{3}{28}}$ . After simplification we find that that $y=\frac{4\sqrt{21}}{3}$ .

In particular, note that by the Pythagorean theorem, $b^2+2^2=y^2$ , hence $b=10\sqrt{3}/3$ . Also, if $C=(c,6)$ , then $y^2=BC^2=4^2+(c-b)^2$ , hence $c-b=8\sqrt{3}/3,$ and thus $c=18\sqrt{3}/3$ . Using similar methods (or symmetry), we determine that $D=(10\sqrt{3}/3,10)$ , $E=(0,8)$ , and $F=(-8\sqrt{3}/3,4)$ . By the Shoelace theorem, $[ABCDEF]=\frac12\left|\begin{array}{cc} 0&0\\ 10\sqrt{3}/3&2\\ 18\sqrt{3}/3&6\\ 10\sqrt{3}/3&10\\ 0&8\\ -8\sqrt{3}/3&4\\ 0&0\\ \end{array}\right|=\frac12|60+180+80-36-60-(-64)|\sqrt{3}/3=48\sqrt{3}.$

Hence the answer is $\boxed{51}$ .

Solution 3

This is similar to solution 2 but faster and easier. First off we see that the y coordinate of F must be 4, the y coordinate of E must be 8, the y coordinate of D must be 10, and the y coordinate of C must be 6 (from the parallel sides of the hexagon). We then use the sine sum angle formula to find the x coordinate of B (lets call it $x$ ): $2\cdot\cos(120)+x\sin(120)=\frac{x\sqrt3}{2}-1=4\rightarrow x=\frac{10\sqrt3}{3}$ . Now that we know $x$ we can find the x coordinate of F in multiple ways, including using the cosine sum angle formula or using the fact that triangle AFE is isosceles and AE is on the y axis. Either way, we find that the x coordinate of F is $-\frac{8\sqrt3}{3}$ . Now, divide ABCDEF into two congruent triangles and a parallelogram: AFE, BCD, and ABDE. The areas of AFE and BCD are each $\frac12\cdot\frac{8\sqrt3}{3}\cdot8=\frac{32\sqrt3}{3}$ . The area of ABDE is $\frac12\cdot8\cdot\frac{10\sqrt3}{3}=\frac{80\sqrt3}3$ . The total area of the hexagon is $2\cdot\frac{32\sqrt3}3+\frac{80\sqrt3}3=\frac{144\sqrt3}{3}=48\sqrt3\rightarrow48+3=\boxed{051}$

Solution 4 (No Trig)

This is also similar to solution 2 but does not use any trig. First we see that the y-coordinates of F, E, D, and C must be 4, 8, and 10, and 6, respectively, as in the previous solution. We can draw a rectangle around hexagon ABCDEF and use negative space to find the area of the hexagon. If we call the distance from the foot of the perpendiculars of B and F to A $x$ and $z$ , respectively, and the distance from the bottom left vertex of the rectangle to the foot of the perpendicular from B $y$ . This tells us that the area of the entire rectangle is $10(x+y+z)$ . Then, if we find the area of the extra triangles and subtract, we find that the area of hexagon ABCDEF as $6x+8z+2y$ . However, noticing that $x=y$ , the area of ABCDEF can also be expressed as $8(x+z)$ . Now we just need to find $x+z$ . Since $AB=AF$ and $\angle BAF = 120$ degrees, $BF=AB\sqrt{3}$ . However, we can find AB by using the Pythagorean Theorem on either of the right triangles formed by dropping perpendiculars from B and F to the x-axis (lets call them ABX and AFY). From triangle ABX we have that $AB=\sqrt{4+x^2}$ , so $BF=\sqrt{3x^2+12}$ . Since AB=AF, we can also form the equation $4+x^2=16+z^2$ . We can also find BF by dropping a perpendicular from B to line FY and using the Pythagorean Theorem on the right triangle formed. This gives us $BF=\sqrt{4+(x+z)^2}$ . Setting our two values of BF equal and substituting $x^2$ as $12+z^2$ and simplifying, we get the equation $3z^4-16z^2-1024=0$ . Now we can use the quadratic formula to get that $z^2=\frac{64}{3}$ or -18, so $z^2=\frac{64}{3}$ . Plugging this value back into the equation $x^2=12+z^2$ , we get that $x^2=\frac{100}{3}$ . Now we get that x+z is $6\sqrt{3}$ , so the area of the hexagon is $8 \cdot 6\sqrt{3}=48\sqrt{3}$ , so the answer is $48+3=\boxed{051}$

~ant08 and sky2025

@@ Line 64: / Line 64: @@
 The total area of the hexagon is <math>2\cdot\frac{32\sqrt3}3+\frac{80\sqrt3}3=\frac{144\sqrt3}{3}=48\sqrt3\rightarrow48+3=\boxed{051}</math>
+==Solution 4 (No Trig)==
+This is also similar to solution 2 but does not use any trig.
+First we see that the y-coordinates of F, E, D, and C must be 4, 8, and 10, and 6, respectively, as in the previous solution. We can draw a rectangle around hexagon ABCDEF and use negative space to find the area of the hexagon. If we call the distance from the foot of the perpendiculars of B and F to A <math>x</math> and <math>z</math>, respectively, and the distance from the bottom left vertex of the rectangle to the foot of the perpendicular from B <math>y</math>. This tells us that the area of the entire rectangle is <math>10(x+y+z)</math>. Then, if we find the area of the extra triangles and subtract, we find that the area of hexagon ABCDEF as <math>6x+8z+2y</math>. However, noticing that <math>x=y</math>, the area of ABCDEF can also be expressed as <math>8(x+z)</math>. Now we just need to find <math>x+z</math>. Since <math>AB=AF</math> and <math>\angle BAF = 120</math> degrees, <math>BF=AB\sqrt{3}</math>. However, we can find AB by using the Pythagorean Theorem on either of the right triangles formed by dropping perpendiculars from B and F to the x-axis (lets call them ABX and AFY).
+From triangle ABX we have that <math>AB=\sqrt{4+x^2}</math>, so <math>BF=\sqrt{3x^2+12}</math>. Since AB=AF, we can also form the equation <math>4+x^2=16+z^2</math>.
+We can also find BF by dropping a perpendicular from B to line FY and using the Pythagorean Theorem on the right triangle formed. This gives us <math>BF=\sqrt{4+(x+z)^2}</math>. Setting our two values of BF equal and substituting <math>x^2</math> as <math>12+z^2</math> and simplifying, we get the equation <math>3z^4-16z^2-1024=0</math>. Now we can use the quadratic formula to get that <math>z^2=\frac{64}{3}</math> or -18, so <math>z^2=\frac{64}{3}</math>. Plugging this value back into the equation <math>x^2=12+z^2</math>, we get that <math>x^2=\frac{100}{3}</math>. Now we get that x+z is <math>6\sqrt{3}</math>, so the area of the hexagon is <math>8 \cdot 6\sqrt{3}=48\sqrt{3}</math>, so the answer is <math>48+3=\boxed{051}</math>
+~ant08 and sky2025
 == See also ==
 {{AIME box|year=2003|n=II|num-b=13|num-a=15}}
 [[Category:Intermediate Geometry Problems]]
 {{MAA Notice}}

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