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Difference between revisions of "2008 AMC 12A Problems/Problem 22"

(→‎Solution 2 (without trigonometry))
(→‎Solution 6 (Answer Choices & Estimation))
 
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== Solution 2 (without trigonometry) ==
 
== Solution 2 (without trigonometry) ==
−
Draw <math>OD</math> and <math>OC</math> as in the diagram. Draw the altitude from <math>O</math> to <math>DC</math> and call the intersection <math>E</math>
+
Draw <math>OD</math> and <math>OC</math> as in the diagram. Draw the altitude from <math>O</math> to <math>DC</math> and call the intersection <math>E</math>.
  
  
Line 118: Line 118:
 
draw((0,0)--(-0.5,3.9686));</asy>
 
draw((0,0)--(-0.5,3.9686));</asy>
  
−
By symmetry, <math>E</math> is the midpoint of <math>DF</math> and <math>OE</math> is an extension of <math>OC</math>. Thus <math>\angle OED = 90^\circ</math>. Since <math>OD = 4</math> and <math>DE = \frac{1}{2}</math>, <math>OE = \sqrt{16-\frac{1}{4}} = \frac{\sqrt{63}}{2} = \frac{3\sqrt{7}}{2}</math>. Since <math>\triangle CED</math> is <math>30-60-90</math>, <math>CE = \frac{\sqrt{3}}{2}</math> (or this can also be deduced from Pythagoras on <math>\triangle CED</math>).
+
By symmetry, <math>E</math> is the midpoint of <math>DF</math> and <math>OE</math> is an extension of <math>OC</math>. Thus <math>\angle OED = 90^\circ</math>. Since <math>OD = 4</math> and <math>DE = \frac{1}{2}</math>, <math>OE = \sqrt{16-\frac{1}{4}} = \frac{\sqrt{63}}{2} = \frac{3\sqrt{7}}{2}</math>. Since <math>\triangle CED</math> is <math>30-60-90</math>, <math>CE = \frac{\sqrt{3}}{2}</math> (this can also be deduced from Pythagoras on <math>\triangle CED</math>).
  
−
Thus <math>OC = \frac{3\sqrt{7}-\sqrt{3}}{2}</math>. As previous solutions noted, <math>\triangle BOC</math> is equilateral, and thus the desired length is <math>x = OC \implies C</math>.
+
Thus <math>OC = \frac{3\sqrt{7}-\sqrt{3}}{2}</math>. As previous solutions noted, <math>\triangle BOC</math> is equilateral, and thus the desired length is <math>x = OC \implies (C)</math>.
  
 
==Solution 3==
 
==Solution 3==
Line 151: Line 151:
 
draw((0.5,-3.9686)--(-0.5,3.9686));</asy>
 
draw((0.5,-3.9686)--(-0.5,3.9686));</asy>
  
−
Looking at the diagram above, we know that <math>BE</math> is a diameter of circle <math>O</math> due to symmetry. Due to Thales' theorem, triangle <math>ABE</math> is a right triangle with <math>A = 90 ^\circ</math>. <math>AE</math> lies on <math>AD</math> and <math>GE</math> because <math>BAD</math> is also a right angle. To find the length of <math>DG</math>, notice that if we draw a line from <math>F</math> to <math>M</math>, the midpoint of line <math>DG</math>, it creates two <math>30</math> - <math>60</math> - <math>90</math> triangles. Therefore, <math>MD = MG = \frac{\sqrt{3}x}{2} \Rightarrow DG = \sqrt{3}x</math>. <math>AE = 2 + \sqrt{3}x</math>
+
Looking at the diagram above, we know that <math>BE</math> is a diameter of circle <math>O</math> due to symmetry. Due to [[Thales' theorem]], triangle <math>ABE</math> is a right triangle with <math>A = 90 ^\circ</math>. <math>AE</math> lies on <math>AD</math> and <math>GE</math> because <math>BAD</math> is also a right angle. To find the length of <math>DG</math>, notice that if we draw a line from <math>F</math> to <math>M</math>, the midpoint of line <math>DG</math>, it creates two <math>30</math> - <math>60</math> - <math>90</math> triangles. Therefore, <math>MD = MG = \frac{\sqrt{3}x}{2} \Rightarrow DG = \sqrt{3}x</math>. <math>AE = 2 + \sqrt{3}x</math>
  
−
Use the Pythagorean theorem on triangle <math>ABE</math>, we get <cmath>(2+\sqrt{3}x)^2 + x^2 = 8^2 \Rightarrow 4 + 3x^2 + 4\sqrt{3}x + x^2 = 64 \Rightarrow x^2 + \sqrt{3}x - 15 = 0</cmath> Using the quadratic formula to solve, we get <cmath>x = \frac{-\sqrt{3} \pm \sqrt{3 -4(1)(-15)}}{2} = \frac{\pm 3\sqrt{7} - \sqrt{3}}{2}</cmath> <math>x</math> must be positive, therefore <cmath>x = \frac{3\sqrt{7} - \sqrt{3}}{2} \Rightarrow C</cmath>
+
Applying the Pythagorean theorem to triangle <math>ABE</math>, we get <cmath>(2+\sqrt{3}x)^2 + x^2 = 8^2 \Rightarrow 4 + 3x^2 + 4\sqrt{3}x + x^2 = 64 \Rightarrow x^2 + \sqrt{3}x - 15 = 0</cmath> Using the quadratic formula to solve, we get <cmath>x = \frac{-\sqrt{3} \pm \sqrt{3 -4(1)(-15)}}{2} = \frac{\pm 3\sqrt{7} - \sqrt{3}}{2}</cmath> <math>x</math> must be positive, therefore <cmath>x = \frac{3\sqrt{7} - \sqrt{3}}{2} \Rightarrow (C)</cmath>
  
 
~Zeric Hang
 
~Zeric Hang
  
−
==Soultion 4 (coordinate bashing)==
+
==Solution 4 (coordinate bashing)==
 
<asy>unitsize(8mm);
 
<asy>unitsize(8mm);
 
defaultpen(linewidth(.8)+fontsize(8));
 
defaultpen(linewidth(.8)+fontsize(8));
Line 171: Line 171:
 
label("\(1\)",(-0.5,3.8),S);</asy>
 
label("\(1\)",(-0.5,3.8),S);</asy>
  
−
We will let <math>O(0,0)</math> be the origin. This way the coordinates of C would be <math>(0,x)</math>. By 30-60-90, the coordinates of D would be <math>(\sqrt{-1}{2}, x + \frac{\sqrt{3}}{2})</math>. The distance <math>(x, y)</math> is from the origin is just <math>\sqrt{x^2 + y^2}</math>. Therefore, the distance D is from the origin is both 4 and <math>\frac{1}{4} + x^2 + x\sqrt{3} + \frac{3}{4} = x^2 + x\sqrt{3} + 1 = 16</math>. We get the equation mentioned in all the previous solution, using the quadratic formula, we get that <math>x = \frac{3\sqrt{7}-\sqrt{3}}{2} \Rightarrow C</math>
+
We will let <math>O(0,0)</math> be the origin. This way the coordinates of <math>C</math> will be <math>(0,y)</math>. By <math>30-60-90</math>, the coordinates of <math>D</math> will be <math>\left(-\frac{1}{2}, y + \frac{\sqrt{3}}{2}\right)</math>. The distance any point with coordinates <math>(x, y)</math> is from the origin is <math>\sqrt{x^2 + y^2}</math>. Therefore, the distance <math>D</math> is from the origin is <math>4</math> and <math>\frac{1}{4} + x^2 + x\sqrt{3} + \frac{3}{4} = x^2 + x\sqrt{3} + 1 = 16</math>. We get the quadratic equation mentioned in solution 2. Using the quadratic formula, we get that <math>x = \frac{3\sqrt{7}-\sqrt{3}}{2} \Rightarrow (C)</math>
 +
 
 +
Note: Since <math>C</math> and <math>D</math> are not labeled in the diagram, refer to solution 1 for the location of points <math>C</math> and <math>D</math>.
  
 
==Solution 5==
 
==Solution 5==
Line 194: Line 196:
 
draw((0.5,3.9686)--(-2.95,2));</asy>
 
draw((0.5,3.9686)--(-2.95,2));</asy>
  
−
Notice that <math>\overarc{AE}</math> is one-sixth the circumference of the circle. Therefore, <math>\overline{AE}</math> is the side length of an inscribed hexagon. The side length of such a hexagon is simply the radius, or <math>4</math>. <math>\angle AFE</math> is a right angle, therefore <math>\triangle AFE</math> is a right triangle. <math>\overline{AF}</math> is half the length of <math>1</math>, or <math>\frac{1}{2}</math>. The length of <math>\overline{EF}</math> is <math>x</math> plus the altitude length of one of the equilateral triangles, or <math>x+\frac{\sqrt{3}}{2}</math>. Using the [[Pythagorean Theorem]], we get
+
Notice that <math>\overarc{AE}</math> is <math>\frac16</math> the circumference of the circle. Therefore, <math>\overline{AE}</math> is the side length of an inscribed hexagon with side length <math>4</math>. <math>\triangle AFE</math> is a right triangle with <math>\overline{AF}=\frac12</math>. The length of <math>\overline{EF}=x+\frac{\sqrt{3}}{2}</math>. Using the [[Pythagorean Theorem]], we get
  
  
Line 200: Line 202:
  
  
−
Solving for <math>x</math>, we get <math>x =</math>
+
Solving for <math>x</math>, we get <math>x = \frac{3\sqrt{7}-\sqrt{3}}{2}\ \boxed{\text{C}}</math>
−
 
 
−
<math>\frac{3\sqrt{7}-\sqrt{3}}{2}</math> , or <math>\boxed{\text{C}}</math>.
 
  
 +
==Solution 6 (Answer Choices & Estimation)==
 +
The smallest distance between the intersection between two adjacent placemats and the circle is a bit less than <math>1</math>. Thus, the answer will be a bit more than <math>4-1=3.</math> Going up, we guess that the next one up is the answer, <math>\frac{3\sqrt{7}-\sqrt{3}}{2},</math> which is about <math>3.1</math>. The next one up after that is <math>2\sqrt{3},</math> which is about <math>3.5</math> and seems too big. So we circle <math>\boxed{\text{(C) }\frac{3\sqrt{7}-\sqrt{3}}{2}}</math> and are happy we just cheesed the <math>AMC</math> <math>10</math> <math>2008</math> problem <math>25.</math>
  
 
==See Also==
 
==See Also==

Latest revision as of 05:37, 20 September 2024

The following problem is from both the 2008 AMC 12A #22 and 2008 AMC 10A #25, so both problems redirect to this page.

Problem

A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$?

[asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.55,2.1),E); label("\(1\)",(-0.5,3.8),S);[/asy]

$\mathrm{(A)}\ 2\sqrt{5}-\sqrt{3}\qquad\mathrm{(B)}\ 3\qquad\mathrm{(C)}\ \frac{3\sqrt{7}-\sqrt{3}}{2}\qquad\mathrm{(D)}\ 2\sqrt{3}\qquad\mathrm{(E)}\ \frac{5+2\sqrt{3}}{2}$

Solution 1 (Trigonometry)

Let one of the mats be $ABCD$, and the center be $O$ as shown:

[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.55,2.1),E); label("\(x\)",(0.03,1.5),E); label("\(A\)",(-3.6,2.5513),E); label("\(B\)",(-3.15,1.35),E); label("\(C\)",(0.05,3.20),E); label("\(D\)",(-0.75,4.15),E); label("\(O\)",(0.00,-0.10),E); label("\(1\)",(-0.1,3.8),S); label("\(4\)",(-0.4,2.2),S); draw((0,0)--(0,3.103)); draw((0,0)--(-2.687,1.5513)); draw((0,0)--(-0.5,3.9686));[/asy]

Since there are $6$ mats, $\Delta BOC$ is equilateral (the hexagon with side length $x$ is regular). So, $BC=CO=x$. Also, $\angle OCD = \angle OCB + \angle BCD = 60^\circ+90^\circ=150^\circ$.

By the Law of Cosines: $4^2=1^2+x^2-2\cdot1\cdot x \cdot \cos(150^\circ) \Rightarrow x^2 + x\sqrt{3} - 15 = 0 \Rightarrow x = \frac{-\sqrt{3}\pm 3\sqrt{7}}{2}$.

Since $x$ must be positive, $x = \frac{3\sqrt{7}-\sqrt{3}}{2} \Rightarrow (C)$.

Solution 2 (without trigonometry)

Draw $OD$ and $OC$ as in the diagram. Draw the altitude from $O$ to $DC$ and call the intersection $E$.


[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=((-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle); draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); pair D = rotate(300)*(-3.687,1.5513); pair C = rotate(300)*(-2.687,1.5513); pair EE = foot((0.00,0.00),D,C); draw(D--EE--(0,0)); label("\(x\)",(-1.55,2.1),E); label("\(x\)",(0.03,1.5),E); label("\(A\)",(-3.6,2.5513),E); label("\(B\)",(-3.15,1.35),E); label("\(C\)",(0.05,3.20),E); label("\(D\)",(-0.75,4.15),E); label("\(O\)",(0.00,-0.10),E); label("\(1\)",(-0.1,3.8),S); label("\(4\)",(-0.4,2.2),S); draw((0,0)--(0,3.103)); draw((0,0)--(-2.687,1.5513)); draw((0,0)--(-0.5,3.9686)); label("\(E\)", EE,SE); [/asy]

As proved in the first solution, $\angle OCD = 150^\circ$. That makes $\Delta OCE$ a $30-60-90$ triangle, so $OE = \frac{x}{2}$ and $CE= \frac{x\sqrt 3}{2}$

Since $\Delta ODE$ is a right triangle, $\left({\frac{x}{2}}\right)^2 + \left({\frac{x\sqrt 3}{2} +1}\right)^2 = 4^2 \Rightarrow x^2+x\sqrt3-15 = 0$

Solving for $x$ gives $x =\frac{3\sqrt{7}-\sqrt{3}}{2}\Rightarrow (C)$

Solution 3 (simply Pythagorean Theorem)

[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.55,2.1),E); label("\(x\)",(0.03,1.5),E); label("\(A\)",(-3.6,2.5513),E); label("\(B\)",(-3.15,1.35),E); label("\(C\)",(0.05,3.20),E); label("\(D\)",(-0.75,4.15),E); label("\(E\)",(0,4.17)); label("\(F\)",(0.75,4.15),W); label("\(O\)",(0.00,-0.10),E); label("\(1\)",(-0.1,3.8),S); label("\(4\)",(-0.4,2.2),S); draw((0,0)--(0,3.103)); draw((0,0)--(-2.687,1.5513)); draw((0,0)--(-0.5,3.9686)); draw((0,0)--(-0.5,3.9686));[/asy]

By symmetry, $E$ is the midpoint of $DF$ and $OE$ is an extension of $OC$. Thus $\angle OED = 90^\circ$. Since $OD = 4$ and $DE = \frac{1}{2}$, $OE = \sqrt{16-\frac{1}{4}} = \frac{\sqrt{63}}{2} = \frac{3\sqrt{7}}{2}$. Since $\triangle CED$ is $30-60-90$, $CE = \frac{\sqrt{3}}{2}$ (this can also be deduced from Pythagoras on $\triangle CED$).

Thus $OC = \frac{3\sqrt{7}-\sqrt{3}}{2}$. As previous solutions noted, $\triangle BOC$ is equilateral, and thus the desired length is $x = OC \implies (C)$.

Solution 3

[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.95,3),E); label("\(A\)",(-3.6,2.5513),E); label("\(C\)",(0.05,3.20),E); label("\(E\)",(0.40,-3.60),E); label("\(B\)",(-0.75,4.15),E); label("\(D\)",(-2.62,1.5),E); label("\(F\)",(-2.64,-1.43),E); label("\(G\)",(-0.2,-2.8),E); label("\( \sqrt{3}x\)",(-1.5,-0.5),E); label("\(M\)",(-2,-0.9),E); label("\(O\)",(0.00,-0.10),E); label("\(1\)",(-2.7,2.3),S); label("\(1\)",(0.1,-3.4),S); label("\(8\)",(-0.3,0),S); draw((0,-3.103)--(-2.687,1.5513)); draw((0.5,-3.9686)--(-0.5,3.9686));[/asy]

Looking at the diagram above, we know that $BE$ is a diameter of circle $O$ due to symmetry. Due to Thales' theorem, triangle $ABE$ is a right triangle with $A = 90 ^\circ$. $AE$ lies on $AD$ and $GE$ because $BAD$ is also a right angle. To find the length of $DG$, notice that if we draw a line from $F$ to $M$, the midpoint of line $DG$, it creates two $30$ - $60$ - $90$ triangles. Therefore, $MD = MG = \frac{\sqrt{3}x}{2} \Rightarrow DG = \sqrt{3}x$. $AE = 2 + \sqrt{3}x$

Applying the Pythagorean theorem to triangle $ABE$, we get \[(2+\sqrt{3}x)^2 + x^2 = 8^2 \Rightarrow 4 + 3x^2 + 4\sqrt{3}x + x^2 = 64 \Rightarrow x^2 + \sqrt{3}x - 15 = 0\] Using the quadratic formula to solve, we get \[x = \frac{-\sqrt{3} \pm \sqrt{3 -4(1)(-15)}}{2} = \frac{\pm 3\sqrt{7} - \sqrt{3}}{2}\] $x$ must be positive, therefore \[x = \frac{3\sqrt{7} - \sqrt{3}}{2} \Rightarrow (C)\]

~Zeric Hang

Solution 4 (coordinate bashing)

[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.55,2.1),E); label("\(1\)",(-0.5,3.8),S);[/asy]

We will let $O(0,0)$ be the origin. This way the coordinates of $C$ will be $(0,y)$. By $30-60-90$, the coordinates of $D$ will be $\left(-\frac{1}{2}, y + \frac{\sqrt{3}}{2}\right)$. The distance any point with coordinates $(x, y)$ is from the origin is $\sqrt{x^2 + y^2}$. Therefore, the distance $D$ is from the origin is $4$ and $\frac{1}{4} + x^2 + x\sqrt{3} + \frac{3}{4} = x^2 + x\sqrt{3} + 1 = 16$. We get the quadratic equation mentioned in solution 2. Using the quadratic formula, we get that $x = \frac{3\sqrt{7}-\sqrt{3}}{2} \Rightarrow (C)$

Note: Since $C$ and $D$ are not labeled in the diagram, refer to solution 1 for the location of points $C$ and $D$.

Solution 5

[asy]unitsize(8mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("\(x\)",(-1.55,2.1),E); label("\(A\)",(-3.6,2.5513),E); label("\(B\)",(-3.15,1.35),E); label("\(C\)",(0.05,3.20),E); label("\(D\)",(-0.75,4.15),E); label("\(E\)",(0.3,4.15),E); label("\(F\)",(-3.4,1.89),E); draw((0.5,3.9686)--(-3.13,2.45)); draw((0.5,3.9686)--(-2.95,2));[/asy]

Notice that $\overarc{AE}$ is $\frac16$ the circumference of the circle. Therefore, $\overline{AE}$ is the side length of an inscribed hexagon with side length $4$. $\triangle AFE$ is a right triangle with $\overline{AF}=\frac12$. The length of $\overline{EF}=x+\frac{\sqrt{3}}{2}$. Using the Pythagorean Theorem, we get


$4^2 = \left(\frac{1}{2}\right)^2 + \left(x+\frac{\sqrt{3}}{2}\right)^2$


Solving for $x$, we get $x = \frac{3\sqrt{7}-\sqrt{3}}{2}\ \boxed{\text{C}}$

Solution 6 (Answer Choices & Estimation)

The smallest distance between the intersection between two adjacent placemats and the circle is a bit less than $1$. Thus, the answer will be a bit more than $4-1=3.$ Going up, we guess that the next one up is the answer, $\frac{3\sqrt{7}-\sqrt{3}}{2},$ which is about $3.1$. The next one up after that is $2\sqrt{3},$ which is about $3.5$ and seems too big. So we circle $\boxed{\text{(C) }\frac{3\sqrt{7}-\sqrt{3}}{2}}$ and are happy we just cheesed the $AMC$ $10$ $2008$ problem $25.$

See Also

2008 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 24 		Followed by
Last question
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
2008 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 21 		Followed by
Problem 23
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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