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Difference between revisions of "2012 AMC 12B Problems/Problem 20"

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== Problem 20 ==
 
== Problem 20 ==
  
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A trapezoid has side lengths 3, 5, 7, and 11. The sums of all the possible areas of the trapezoid can be written in the form of <math>r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3</math>, where <math>r_1</math>, <math>r_2</math>, and <math>r_3</math> are rational numbers and <math>n_1</math> and <math>n_2</math> are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to <math>r_1+r_2+r_3+n_1+n_2</math>?
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A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of <math>r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3</math>, where <math>r_1</math>, <math>r_2</math>, and <math>r_3</math> are rational numbers and <math>n_1</math> and <math>n_2</math> are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to <math>r_1+r_2+r_3+n_1+n_2</math>?
  
 
<math>\textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65</math>
 
<math>\textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65</math>
  
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==Solution==
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==Solution 1==
 
Name the trapezoid <math>ABCD</math>, where <math>AB</math> is parallel to <math>CD</math>, <math>AB<CD</math>, and <math>AD<BC</math>. Draw a line through <math>B</math> parallel to <math>AD</math>, crossing the side <math>CD</math> at <math>E</math>. Then <math>BE=AD</math>, <math>EC=DC-DE=DC-AB</math>. One needs to guarantee that <math>BE+EC>BC</math>, so there are only three possible trapezoids:
 
Name the trapezoid <math>ABCD</math>, where <math>AB</math> is parallel to <math>CD</math>, <math>AB<CD</math>, and <math>AD<BC</math>. Draw a line through <math>B</math> parallel to <math>AD</math>, crossing the side <math>CD</math> at <math>E</math>. Then <math>BE=AD</math>, <math>EC=DC-DE=DC-AB</math>. One needs to guarantee that <math>BE+EC>BC</math>, so there are only three possible trapezoids:
  
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In the first case, <math>\cos(\angle BCD) = (8^2+7^2-5^2)/(2\cdot 7\cdot 8) = 11/14</math>, so <math>\sin (\angle BCD) = \sqrt{1-121/196} = 5\sqrt{3}/14</math>. Therefore the area of this trapezoid is <math>\frac{1}{2} (3+11) \cdot 7 \cdot 5\sqrt{3}/14 = \frac{35}{2}\sqrt{3}</math>.
+
In the first case, by Law of Cosines, <math>\cos(\angle BCD) = (8^2+7^2-5^2)/(2\cdot 7\cdot 8) = 11/14</math>, so <math>\sin (\angle BCD) = \sqrt{1-121/196} = 5\sqrt{3}/14</math>. Therefore the area of this trapezoid is <math>\frac{1}{2} (3+11) \cdot 7 \cdot 5\sqrt{3}/14 = \frac{35}{2}\sqrt{3}</math>.
  
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In the first case, <math>\cos(\angle BCD) = (6^2+7^2-3^2)/(2\cdot 6\cdot 7) = 19/21</math>, so <math>\sin (\angle BCD) = \sqrt{1-361/441} = 4\sqrt{5}/21</math>. Therefore the area of this trapezoid is <math>\frac{1}{2} (5+11) \cdot 7 \cdot 4\sqrt{5}/21 =\frac{32}{3}\sqrt{5}</math>.
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In the second case, <math>\cos(\angle BCD) = (6^2+7^2-3^2)/(2\cdot 6\cdot 7) = 19/21</math>, so <math>\sin (\angle BCD) = \sqrt{1-361/441} = 4\sqrt{5}/21</math>. Therefore the area of this trapezoid is <math>\frac{1}{2} (5+11) \cdot 7 \cdot 4\sqrt{5}/21 =\frac{32}{3}\sqrt{5}</math>.
 +
 
 +
In the third case, <math>\angle BCD = 90^{\circ}</math>, therefore the area of this trapezoid is <math>\frac{1}{2} (7+11) \cdot 3 = 27</math>.
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 +
So <math>r_1 + r_2 + r_3 + n_1 + n_2 = 17.5 + 10.666... + 27 + 3 + 5</math>, which rounds down to <math>\boxed{\textbf{(D)}\ 63}</math>.
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==Solution 2==
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[[File:2012AMC12BProblem20Solution2.PNG|center|500px]]
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Let the area of the trapezoid be <math>S</math>, the area of the triangle be <math>S_1</math>, the area of the parallelogram be <math>S_2</math>.
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By [https://en.wikipedia.org/wiki/Heron%27s_formula Heron's Formula] <math>S_1 = \sqrt{\frac{b+c+d-a}{2} \cdot \frac{c+d-a-b}{2} \cdot \frac{a+b+d-c}{2} \cdot \frac{b+c-a-d}{2}}</math>
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 +
<math>S_2 = \frac{S_1 \cdot 2}{c-a} \cdot a = \frac{2aS_1}{c-a}</math>
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<math>S = S_1 + S_2 = S_1(1+\frac{2a}{c-a}) = S_1 \cdot \frac{c+a}{c-a} = \frac14 \cdot \frac{c+a}{c-a} \cdot \sqrt{(b+c+d-a)(c+d-a-b)(a+b+d-c)(b+c-a-d)}</math>
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If <math>a = 3</math>, <math>b = 7</math>, <math>c = 11</math>, <math>d = 5</math>
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<math>S = \frac14 \cdot \frac{14}{8} \cdot \sqrt{(7+11+5-3)(11+5-3-7)(3+7+5-11)(7+11-3-5)} = \frac{35\sqrt{3}}{2}</math>
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 +
 
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If <math>a = 3</math>, <math>b = 11</math>, <math>c = 5</math>, <math>d = 7</math>
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<math>S = \frac14 \cdot \frac{8}{2} \cdot \sqrt{(11+5+7-3)(5+7-3-11)(3+5+7-11)(11+5-3-7)}</math>, which is impossible as <math>5+7-3-11 = -2</math>
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 +
 
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If <math>a = 3</math>, <math>b = 5</math>, <math>c = 7</math>, <math>d = 11</math>
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<math>S = \frac14 \cdot \frac{10}{4} \cdot \sqrt{(5+7+11-3)(7+11-3-5)(3+5+11-7)(5+7-11-3)}</math>, which is impossible as <math>5+7-11-3 = -2</math>
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 +
 
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If <math>a = 5</math>, <math>b = 11</math>, <math>c = 7</math>, <math>d = 3</math>
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<math>S = \frac14 \cdot \frac{12}{2} \cdot \sqrt{(11+7+3-5)(7+3-5-11)(5+11+3-7)(11+7-5-3)}</math>, which is impossible as <math>7+3-5-11 = -6</math>
 +
 
 +
 
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If <math>a = 5</math>, <math>b = 3</math>, <math>c = 11</math>, <math>d = 7</math>
 +
 
 +
<math>S = \frac14 \cdot \frac{16}{6} \cdot \sqrt{(3+11+7-5)(11+7-5-3)(5+3+7-11)(3+11-5-7)} = \frac{32\sqrt{5}}{3}</math>
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If <math>a = 7</math>, <math>b = 3</math>, <math>c = 11</math>, <math>d = 5</math>
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<math>S = \frac14 \cdot \frac{18}{4} \cdot \sqrt{(3+11+5-7)(11+5-7-)(7+5+5-11)(3+11-7-5)} = 27</math>
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Thus the answer is <math>\frac{35}{2} + \frac{32}{3} + 27 + 3 + 5</math>, which rounds down to <math>\boxed{\textbf{(D) } 63}</math>
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~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
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Note: the three invalid cases can also be determined by the triangle inequality.
 +
 
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== Video Solution ==
 +
 
 +
https://youtu.be/8w1vrsD2urs  ~Math Problem Solving Skills
  
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In the first case, <math>\angle BCD = 90^{\circ}</math>, therefore the area of this trapezoid is <math>\frac{1}{2} (7+11) \cdot 3 = 27</math>.
 
  
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So <math>r_1 + r_2 + r_3 + n_1 + n_2 = 17.5 + 10.666... + 27 + 3 + 5</math>, which is rounded down to <math>63... \framebox{D}</math>.
 
  
 
== See Also ==
 
== See Also ==

Latest revision as of 07:03, 6 September 2024

Problem 20

A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$?

$\textbf{(A)}\ 57\qquad\textbf{(B)}\ 59\qquad\textbf{(C)}\ 61\qquad\textbf{(D)}\ 63\qquad\textbf{(E)}\ 65$

Solution 1

Name the trapezoid $ABCD$, where $AB$ is parallel to $CD$, $AB<CD$, and $AD<BC$. Draw a line through $B$ parallel to $AD$, crossing the side $CD$ at $E$. Then $BE=AD$, $EC=DC-DE=DC-AB$. One needs to guarantee that $BE+EC>BC$, so there are only three possible trapezoids:

\[AB=3, BC=7, CD=11, DA=5, CE=8\] \[AB=5, BC=7, CD=11, DA=3, CE=6\] \[AB=7, BC=5, CD=11, DA=3, CE=4\]


In the first case, by Law of Cosines, $\cos(\angle BCD) = (8^2+7^2-5^2)/(2\cdot 7\cdot 8) = 11/14$, so $\sin (\angle BCD) = \sqrt{1-121/196} = 5\sqrt{3}/14$. Therefore the area of this trapezoid is $\frac{1}{2} (3+11) \cdot 7 \cdot 5\sqrt{3}/14 = \frac{35}{2}\sqrt{3}$.

In the second case, $\cos(\angle BCD) = (6^2+7^2-3^2)/(2\cdot 6\cdot 7) = 19/21$, so $\sin (\angle BCD) = \sqrt{1-361/441} = 4\sqrt{5}/21$. Therefore the area of this trapezoid is $\frac{1}{2} (5+11) \cdot 7 \cdot 4\sqrt{5}/21 =\frac{32}{3}\sqrt{5}$.

In the third case, $\angle BCD = 90^{\circ}$, therefore the area of this trapezoid is $\frac{1}{2} (7+11) \cdot 3 = 27$.

So $r_1 + r_2 + r_3 + n_1 + n_2 = 17.5 + 10.666... + 27 + 3 + 5$, which rounds down to $\boxed{\textbf{(D)}\ 63}$.

Solution 2

2012AMC12BProblem20Solution2.PNG

Let the area of the trapezoid be $S$, the area of the triangle be $S_1$, the area of the parallelogram be $S_2$.

By Heron's Formula $S_1 = \sqrt{\frac{b+c+d-a}{2} \cdot \frac{c+d-a-b}{2} \cdot \frac{a+b+d-c}{2} \cdot \frac{b+c-a-d}{2}}$

$S_2 = \frac{S_1 \cdot 2}{c-a} \cdot a = \frac{2aS_1}{c-a}$

$S = S_1 + S_2 = S_1(1+\frac{2a}{c-a}) = S_1 \cdot \frac{c+a}{c-a} = \frac14 \cdot \frac{c+a}{c-a} \cdot \sqrt{(b+c+d-a)(c+d-a-b)(a+b+d-c)(b+c-a-d)}$

If $a = 3$, $b = 7$, $c = 11$, $d = 5$

$S = \frac14 \cdot \frac{14}{8} \cdot \sqrt{(7+11+5-3)(11+5-3-7)(3+7+5-11)(7+11-3-5)} = \frac{35\sqrt{3}}{2}$


If $a = 3$, $b = 11$, $c = 5$, $d = 7$

$S = \frac14 \cdot \frac{8}{2} \cdot \sqrt{(11+5+7-3)(5+7-3-11)(3+5+7-11)(11+5-3-7)}$, which is impossible as $5+7-3-11 = -2$


If $a = 3$, $b = 5$, $c = 7$, $d = 11$

$S = \frac14 \cdot \frac{10}{4} \cdot \sqrt{(5+7+11-3)(7+11-3-5)(3+5+11-7)(5+7-11-3)}$, which is impossible as $5+7-11-3 = -2$


If $a = 5$, $b = 11$, $c = 7$, $d = 3$

$S = \frac14 \cdot \frac{12}{2} \cdot \sqrt{(11+7+3-5)(7+3-5-11)(5+11+3-7)(11+7-5-3)}$, which is impossible as $7+3-5-11 = -6$


If $a = 5$, $b = 3$, $c = 11$, $d = 7$

$S = \frac14 \cdot \frac{16}{6} \cdot \sqrt{(3+11+7-5)(11+7-5-3)(5+3+7-11)(3+11-5-7)} = \frac{32\sqrt{5}}{3}$


If $a = 7$, $b = 3$, $c = 11$, $d = 5$

$S = \frac14 \cdot \frac{18}{4} \cdot \sqrt{(3+11+5-7)(11+5-7-)(7+5+5-11)(3+11-7-5)} = 27$

Thus the answer is $\frac{35}{2} + \frac{32}{3} + 27 + 3 + 5$, which rounds down to $\boxed{\textbf{(D) } 63}$

~isabelchen

Note: the three invalid cases can also be determined by the triangle inequality.

Video Solution

https://youtu.be/8w1vrsD2urs ~Math Problem Solving Skills


See Also

2012 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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All AMC 12 Problems and Solutions

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