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Difference between revisions of "2005 AIME I Problems/Problem 7"
m (→‎Solution)
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== Solution ==
 
== Solution ==
 
=== Solution 1 ===
 
=== Solution 1 ===
 +
<asy>draw((0,0)--(20.87,0)--(15.87,8.66)--(5,8.66)--cycle);
 +
draw((5,8.66)--(5,0));
 +
draw((15.87,8.66)--(15.87,0));
 +
draw((5,8.66)--(16.87,6.928));
 +
label("$A$",(0,0),SW);
 +
label("$B$",(20.87,0),SE);
 +
label("$E$",(15.87,8.66),NE);
 +
label("$D$",(5,8.66),NW);
 +
label("$P$",(5,0),S);
 +
label("$Q$",(15.87,0),S);
 +
label("$C$",(16.87,7),E);
 +
label("$12$",(10.935,7.794),S);
 +
label("$10$",(2.5,4.5),W);
 +
label("$10$",(18.37,4.5),E);
 +
</asy>
 +
=== Solution 2 ===
 
<center>[[Image:AIME_2005I_Solution_7_1.png]]</center>
 
<center>[[Image:AIME_2005I_Solution_7_1.png]]</center>
  
 
Draw the [[perpendicular]]s from <math>C</math> and <math>D</math> to <math>AB</math>, labeling the intersection points as <math>E</math> and <math>F</math>. This forms 2 <math>30-60-90</math> [[right triangle]]s, so <math>AE = 5</math> and <math>BF = 4</math>. Also, if we draw the horizontal line extending from <math>C</math> to a point <math>G</math> on the line <math>DE</math>, we find another right triangle <math>\triangle DGC</math>. <math>DG = DE - CF = 5\sqrt{3} - 4\sqrt{3} = \sqrt{3}</math>. The [[Pythagorean Theorem]] yields that <math>GC^2 = 12^2 - \sqrt{3}^2 = 141</math>, so <math>EF = GC = \sqrt{141}</math>. Therefore, <math>AB = 5 + 4 + \sqrt{141} = 9 + \sqrt{141}</math>, and <math>p + q = \boxed{150}</math>.
 
Draw the [[perpendicular]]s from <math>C</math> and <math>D</math> to <math>AB</math>, labeling the intersection points as <math>E</math> and <math>F</math>. This forms 2 <math>30-60-90</math> [[right triangle]]s, so <math>AE = 5</math> and <math>BF = 4</math>. Also, if we draw the horizontal line extending from <math>C</math> to a point <math>G</math> on the line <math>DE</math>, we find another right triangle <math>\triangle DGC</math>. <math>DG = DE - CF = 5\sqrt{3} - 4\sqrt{3} = \sqrt{3}</math>. The [[Pythagorean Theorem]] yields that <math>GC^2 = 12^2 - \sqrt{3}^2 = 141</math>, so <math>EF = GC = \sqrt{141}</math>. Therefore, <math>AB = 5 + 4 + \sqrt{141} = 9 + \sqrt{141}</math>, and <math>p + q = \boxed{150}</math>.
  
−
=== Solution 2 ===
+
=== Solution 3 ===
 
<center>[[Image:AIME_2005I_Solution_7_2.png]]</center>
 
<center>[[Image:AIME_2005I_Solution_7_2.png]]</center>
 
Extend <math>AD</math> and <math>BC</math> to an intersection at point <math>E</math>. We get an [[equilateral triangle]] <math>ABE</math>. We denote the length of a side of <math>\triangle ABE</math> as <math>s</math> and solve for it using the [[Law of Cosines]]: <cmath>12^2 = (s - 10)^2 + (s - 8)^2 - 2(s - 10)(s - 8)\cos{60}</cmath>
 
Extend <math>AD</math> and <math>BC</math> to an intersection at point <math>E</math>. We get an [[equilateral triangle]] <math>ABE</math>. We denote the length of a side of <math>\triangle ABE</math> as <math>s</math> and solve for it using the [[Law of Cosines]]: <cmath>12^2 = (s - 10)^2 + (s - 8)^2 - 2(s - 10)(s - 8)\cos{60}</cmath>
 
<cmath>144 = 2s^2 - 36s + 164 - (s^2 - 18s + 80)</cmath> This simplifies to <math>s^2 - 18s - 60=0</math>; the [[quadratic formula]] yields the (discard the negative result) same result of <math>9 + \sqrt{141}</math>.
 
<cmath>144 = 2s^2 - 36s + 164 - (s^2 - 18s + 80)</cmath> This simplifies to <math>s^2 - 18s - 60=0</math>; the [[quadratic formula]] yields the (discard the negative result) same result of <math>9 + \sqrt{141}</math>.
  
−
=== Solution 3 ===
+
=== Solution 4 ===
 
Extend <math>BC</math> and <math>AD</math> to meet at point <math>E</math>, forming an equilateral triangle <math>\triangle ABE</math>. Draw a line from <math>C</math> parallel to <math>AB</math> so that it intersects <math>AD</math> at point <math>F</math>. Then, apply [[Stewart's Theorem]] on <math>\triangle CFE</math>. Let <math>CE=x</math>. <cmath>2x(x-2) + 12^2x = 2x^2 + x^2(x-2)</cmath> <cmath>x^3 - 2x^2 - 140x = 0</cmath> By the quadratic formula (discarding the negative result), <math>x = 1 + \sqrt{141}</math>, giving <math>AB = 9 + \sqrt{141}</math> for a final answer of <math>p+q=150</math>.
 
Extend <math>BC</math> and <math>AD</math> to meet at point <math>E</math>, forming an equilateral triangle <math>\triangle ABE</math>. Draw a line from <math>C</math> parallel to <math>AB</math> so that it intersects <math>AD</math> at point <math>F</math>. Then, apply [[Stewart's Theorem]] on <math>\triangle CFE</math>. Let <math>CE=x</math>. <cmath>2x(x-2) + 12^2x = 2x^2 + x^2(x-2)</cmath> <cmath>x^3 - 2x^2 - 140x = 0</cmath> By the quadratic formula (discarding the negative result), <math>x = 1 + \sqrt{141}</math>, giving <math>AB = 9 + \sqrt{141}</math> for a final answer of <math>p+q=150</math>.
  

Revision as of 19:12, 3 March 2020

Problem

In quadrilateral $ABCD,\ BC=8,\ CD=12,\ AD=10,$ and $m\angle A= m\angle B = 60^\circ.$ Given that $AB = p + \sqrt{q},$ where $p$ and $q$ are positive integers, find $p+q.$

Solution

Solution 1

[asy]draw((0,0)--(20.87,0)--(15.87,8.66)--(5,8.66)--cycle); draw((5,8.66)--(5,0)); draw((15.87,8.66)--(15.87,0)); draw((5,8.66)--(16.87,6.928)); label("$A$",(0,0),SW); label("$B$",(20.87,0),SE); label("$E$",(15.87,8.66),NE); label("$D$",(5,8.66),NW); label("$P$",(5,0),S); label("$Q$",(15.87,0),S); label("$C$",(16.87,7),E); label("$12$",(10.935,7.794),S); label("$10$",(2.5,4.5),W); label("$10$",(18.37,4.5),E); [/asy]

Solution 2

AIME 2005I Solution 7 1.png

Draw the perpendiculars from $C$ and $D$ to $AB$, labeling the intersection points as $E$ and $F$. This forms 2 $30-60-90$ right triangles, so $AE = 5$ and $BF = 4$. Also, if we draw the horizontal line extending from $C$ to a point $G$ on the line $DE$, we find another right triangle $\triangle DGC$. $DG = DE - CF = 5\sqrt{3} - 4\sqrt{3} = \sqrt{3}$. The Pythagorean Theorem yields that $GC^2 = 12^2 - \sqrt{3}^2 = 141$, so $EF = GC = \sqrt{141}$. Therefore, $AB = 5 + 4 + \sqrt{141} = 9 + \sqrt{141}$, and $p + q = \boxed{150}$.

Solution 3

AIME 2005I Solution 7 2.png

Extend $AD$ and $BC$ to an intersection at point $E$. We get an equilateral triangle $ABE$. We denote the length of a side of $\triangle ABE$ as $s$ and solve for it using the Law of Cosines: \[12^2 = (s - 10)^2 + (s - 8)^2 - 2(s - 10)(s - 8)\cos{60}\] \[144 = 2s^2 - 36s + 164 - (s^2 - 18s + 80)\] This simplifies to $s^2 - 18s - 60=0$; the quadratic formula yields the (discard the negative result) same result of $9 + \sqrt{141}$.

Solution 4

Extend $BC$ and $AD$ to meet at point $E$, forming an equilateral triangle $\triangle ABE$. Draw a line from $C$ parallel to $AB$ so that it intersects $AD$ at point $F$. Then, apply Stewart's Theorem on $\triangle CFE$. Let $CE=x$. \[2x(x-2) + 12^2x = 2x^2 + x^2(x-2)\] \[x^3 - 2x^2 - 140x = 0\] By the quadratic formula (discarding the negative result), $x = 1 + \sqrt{141}$, giving $AB = 9 + \sqrt{141}$ for a final answer of $p+q=150$.

See also

2005 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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