2014 AIME II Problems/Problem 14

Revision as of 16:31, 14 February 2016 by FractalMathHistory (talk | contribs) (Solution)

Problem

In $\triangle{ABC}, AB=10, \angle{A}=30^\circ$ , and $\angle{C=45^\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Diagram

http://data.artofproblemsolving.com/images/wiki/5/59/AOPS_wiki.PNG ( This is the diagram.)

Solution

As we can see,


$M$ is the midpoint of $BC$ and $N$ is the midpoint of $HM$


$AHC$ is a $45-45-90$ triangle, so $\angle{HAB}=15^\circ$.


$AHD$ is $30-60-90$ triangle.


$AH$ and $PN$ are parallel lines so $PND$ is $30-60-90$ triangle also.


Then if we use those informations we get $AD=2HD$ and


$PD=2ND$ and $AP=AD-PD=2HD-2ND=2HN$ or $AP=2HN=HM$


Now we know that $HM=AP$, we can find for $HM$ which is simpler to find.


We can use point $B$ to split it up as $HM=HB+BM$,


We can chase those lengths and we would get


$AB=10$, so $OB=5$, so $BC=5\sqrt{2}$, so $BM=\dfrac{1}{2} \cdot BC=\dfrac{5\sqrt{2}}{2}$


We can also use Law of Sines:

\[\frac{BC}{AB}=\frac{\sin\angle A}{\sin\angle C}\] \[\frac{BC}{10}=\frac{\frac{1}{2}}{\frac{\sqrt{2}}{2}}\implies BC=5\sqrt{2}\]

Then using right triangle $AHB$, we have $HB=10 sin 15^\circ$


So $HB=10 sin 15^\circ=\dfrac{5(\sqrt{6}-\sqrt{2})}{2}$.


And we know that $AP = HM = HB + BM = \frac{5(\sqrt6-\sqrt2)}{2} + \frac{5\sqrt2}{2} = \frac{5\sqrt6}{2}$.


Finally if we calculate $(AP)^2$.


$(AP)^2=\dfrac{150}{4}=\dfrac{75}{2}$. So our final answer is $75+2=77$.


$m+n=\boxed{077}$


Thank you.


-Gamjawon

Solution 2. Here's a solution that doesn't need sin 15^\circ.

As above, get to $AP=HM$. As in the figure, let $O$ be the foot of the perpendicular from $B$ to $AC$. Then $BCO$ is a 45-45-90 triangle, and $ABO$ is a 30-60-90 triangle. So $BO=5$ and $AO=5\sqrt{3}$; also, $CO=5$, $BC=5\sqrt2$, and $MC=\dfrac{BC}{2}=5\dfrac{\sqrt2}{2}$. But $MO$ and $AH$ are parallel, both being orthogonal to $BC$. Therefore $MH:AO=MC:CO$, or $MH=\dfrac{5\sqrt3}{\sqrt2}$, and we're done.

See also

2014 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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