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Difference between revisions of "Mock AIME 2 2006-2007 Problems/Problem 12"

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== Problem ==
 
== Problem ==
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In quadrilateral <math>\displaystyle ABCD,</math> <math>\displaystyle m \angle DAC= m\angle DBC </math> and <math>\displaystyle \frac{[ADB]}{[ABC]}=\frac12.</math> If <math>\displaystyle AD=4,</math> <math>\displaystyle BC=6</math>, <math>\displaystyle BO=1,</math> and the area of <math>\displaystyle ABCD</math> is <math>\displaystyle \frac{a\sqrt{b}}{c},</math> where <math>\displaystyle a,b,c</math> are relatively prime positive integers, find <math>\displaystyle a+b+c.</math>
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In [[quadrilateral]] <math>ABCD,</math> <math>m \angle DAC= m\angle DBC </math> and <math>\frac{[ADB]}{[ABC]}=\frac12.</math> <math>O</math> is defined to be the intersection of the diagonals of <math>ABCD</math>. If <math>AD=4,</math> <math>BC=6</math>, <math>BO=1,</math> and the [[area]] of <math>ABCD</math> is <math>\frac{a\sqrt{b}}{c},</math> where <math>a,b,c</math> are [[relatively prime]] [[positive integer]]s, find <math>a+b+c.</math>
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Note*: <math>[ABC]</math> and <math>[ADB]</math> refer to the areas of [[triangle]]s <math>ABC</math> and <math>ADB.</math>
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==Solution==
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<math>m\angle DAC=m\angle DBC \Rightarrow ABCD</math> is a cylic quadrilateral.
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Let <math>DO=a, AO=b</math>
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<math>\triangle AOD</math> ~ <math>\triangle BOC \Rightarrow b=\frac{2}{3}</math>
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Also, from the Power of a Point Theorem, <math>DO \cdot BO=AO\cdot CO\Rightarrow CO=\frac{3a}{2}</math>
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Notice <math>\frac{[AOD]}{[BOC]}=(\frac{2}{3})^2\Rightarrow [BOC]=\frac{9}{4}[AOD]</math>
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It is given <math>\frac{[AOD]+[AOB]}{[AOB]+[BOC]}=\frac{[ADB]}{[ABC]}=\frac{1}{2} \Rightarrow [AOB]=\frac{[AOD]}{4}</math>
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Note that <math>\sin{\angle AOB}=\sin{(180-\angle AOD)}=\sin{\angle AOD}</math>
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Then <math>[AOB]=\frac{\frac{2}{3}\cdot 1\cdot\sin{\angle AOB}}{2}=\frac{\sin{\angle AOD}}{3}</math>
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and <math>[AOD]=\frac{\frac{2}{3}\cdot a\cdot\sin{\angle AOD}}{2}=\frac{a\sin{\angle AOD}}{3}</math>
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<math>\Rightarrow a=4</math>
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<math>[COD]=9[AOD]</math>
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Thus we need to find <math>[ABCD]=\frac{25}{2}[AOD]</math>
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Note that <math>\triangle AOD</math> is isosceles with sides <math>4, 4, \frac{2}{3}</math> so we can draw the altitude from D to split it to two right triangles.
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<math>[AOD]=\frac{\sqrt{143}}{9}</math>
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Thus <math>[ABCD]=\frac{25\sqrt{143}}{18}\rightarrow\boxed{186}</math>
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==See Also==
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{{Mock AIME box|year=2006-2007|n=2|num-b=11|num-a=13}}
  
  
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Note*: <math>\displaystyle[ABC]</math> and <math>\displaystyle[ADB]</math> refer to the areas of triangles <math>\displaystyle ABC</math> and <math>\displaystyle ADB.</math>
 
  
 
== Problem Source ==
 
== Problem Source ==
 
AoPS users 4everwise and Altheman collaborated to create this problem.
 
AoPS users 4everwise and Altheman collaborated to create this problem.

Latest revision as of 09:53, 4 April 2012

Problem

In quadrilateral $ABCD,$ $m \angle DAC= m\angle DBC$ and $\frac{[ADB]}{[ABC]}=\frac12.$ $O$ is defined to be the intersection of the diagonals of $ABCD$. If $AD=4,$ $BC=6$, $BO=1,$ and the area of $ABCD$ is $\frac{a\sqrt{b}}{c},$ where $a,b,c$ are relatively prime positive integers, find $a+b+c.$


Note*: $[ABC]$ and $[ADB]$ refer to the areas of triangles $ABC$ and $ADB.$

Solution

$m\angle DAC=m\angle DBC \Rightarrow ABCD$ is a cylic quadrilateral.

Let $DO=a, AO=b$

$\triangle AOD$ ~ $\triangle BOC \Rightarrow b=\frac{2}{3}$

Also, from the Power of a Point Theorem, $DO \cdot BO=AO\cdot CO\Rightarrow CO=\frac{3a}{2}$

Notice $\frac{[AOD]}{[BOC]}=(\frac{2}{3})^2\Rightarrow [BOC]=\frac{9}{4}[AOD]$

It is given $\frac{[AOD]+[AOB]}{[AOB]+[BOC]}=\frac{[ADB]}{[ABC]}=\frac{1}{2} \Rightarrow [AOB]=\frac{[AOD]}{4}$

Note that $\sin{\angle AOB}=\sin{(180-\angle AOD)}=\sin{\angle AOD}$

Then $[AOB]=\frac{\frac{2}{3}\cdot 1\cdot\sin{\angle AOB}}{2}=\frac{\sin{\angle AOD}}{3}$ and $[AOD]=\frac{\frac{2}{3}\cdot a\cdot\sin{\angle AOD}}{2}=\frac{a\sin{\angle AOD}}{3}$

$\Rightarrow a=4$


$[COD]=9[AOD]$

Thus we need to find $[ABCD]=\frac{25}{2}[AOD]$

Note that $\triangle AOD$ is isosceles with sides $4, 4, \frac{2}{3}$ so we can draw the altitude from D to split it to two right triangles.

$[AOD]=\frac{\sqrt{143}}{9}$

Thus $[ABCD]=\frac{25\sqrt{143}}{18}\rightarrow\boxed{186}$

See Also

Mock AIME 2 2006-2007 (Problems, Source)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15


Problem Source

AoPS users 4everwise and Altheman collaborated to create this problem.

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