Difference between revisions of "1999 AIME Problems/Problem 14"

Revision as of 21:46, 18 October 2007

Problem

Point $P_{}$ is located inside traingle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

Solution

File:1999 AIME-14.png

Solution 1

Drop perpendiculars from $P$ to the three sides of $\triangle ABC$ and let them meet $\overline{AB}, \overline{BC},$ and $\overline{CA}$ at $D, E,$ and $F$ respectively. Let $BE = x, CF = y,$ and $AD = z$ . We have that $\begin{eqnarray*} DP & = & z\tan \theta \\ EP & = & x\tan \theta \\ FP & = & y\tan \theta \end{eqnarray*}$ We can then use the tool of calculating area in two ways $\begin{eqnarray*} [ABC] & = & [PAB] + [PBC] + [PCA] \\ & = & \frac 12 (13)(z\tan \theta) + \frac 12 (14)(x\tan\theta) + \frac 12 (15)(y\tan\theta) \\ & = & \frac 12 \tan\theta(13z + 14x + 15y) \end{eqnarray*}$ On the other hand $\begin{eqnarray*} [ABC] & = & \sqrt {s(s - a)(s - b)(s - c)} \\ & = & \sqrt {21\cdot 6\cdot 7\cdot 8} \\ & = & 84 \end{eqnarray*}$ We still need $13z + 14x + 15y$ though. We have all these right triangles and we haven't even touched Pythagoras. So we give it a shot

\begin{eqnarray} x^2 + x^2\tan^2\theta & = & z^2\tan^2\theta + (13 - z)^2 \\
z^2 + z^2\tan^2\theta & = & y^2\tan^2\theta + (15 - y)^2 \\
y^2 + y^2\tan^2\theta & = & x^2\tan^2\theta + (14 - x)^2 \\
\end{eqnarray*} (Error compiling LaTeX. Unknown error_msg)

But then $(1) + (2) + (3)$ gives $\begin{eqnarray*} x^2 + y^2 + z^2 & = & (14 - x)^2 + (15 - y)^2 + (13 - z)^2 \\ \Rightarrow 13z + 14x + 15y & = & 295 \end{eqnarray*}$ Recall that we found that $[ABC] = \frac 12 \tan\theta(13z + 14x + 15y) = 84$ . Plugging in $13z + 14x + 15y = 295$ we get $\tan \theta = \frac {168}{295}$ giving us $\boxed{463}$ for an answer.

Solution 2

Let $AB = c, BC = a, AC = b, PA = x, PB = y, PC = z$ .

So by the Law of Cosines, we have: $x^2 = z^2 + b^2 - 2bz\cos{\theta}$ $y^2 = x^2 + c^2 - 2cx\cos{\theta}$ $z^2 = y^2 + a^2 - 2ay\cos{\theta}$

Adding these equations and rearranging, we have: $a^2 + b^2 + c^2 = (2bz + 2cx + 2ay)\cos{\theta}$ (1)

Now $[CAP] + [ABP] + [BCP] = [ABC] = \sqrt {(21)(8)(7)(6)} = 84$ , by Heron's Formula.

Now the area of a triangle, $[A] = \frac {mn\sin{\beta}}{2}$ , where $m$ and $n$ are sides on either side of an angle, $\beta$ . So, $[CAP] = \frac {bz\sin{\theta}}{2}$ . $[ABP] = \frac {cx\sin{\theta}}{2}$ . $[BCP] = \frac {ay\sin{\theta}}{2}$

So adding these equations yields: $[ABC] = 84 = \frac {(bz + cx + ay)\sin{\theta}}{2}$ $\Rightarrow 168 = (bz + cx + ay)\sin{\theta}$ (2)

Dividing (2) by (1), we have: $\frac {168}{a^2 + b^2 + c^2} = \frac {(bz + cx + ay)\sin{\theta}}{(2bz + 2cx + 2ay)\cos{\theta}}$ $\Rightarrow \tan{\theta} = \frac {336}{a^2 + b^2 + c^2} = \frac {336}{14^2 + 15^2 + 13^2} = \frac {336}{590} = \frac {168}{295}$

So $m + n = 168 + 295 = \boxed{463}$ .

@@ Line 1: / Line 1: @@
 == Problem ==
-Point <math>\displaystyle P_{}</math> is located inside traingle <math>\displaystyle ABC</math> so that angles <math>\displaystyle PAB, PBC,</math> and <math>\displaystyle PCA</math> are all congruent.  The sides of the triangle have lengths <math>\displaystyle AB=13, BC=14,</math> and <math>\displaystyle CA=15,</math> and the tangent of angle <math>\displaystyle PAB</math> is <math>\displaystyle m/n,</math> where <math>\displaystyle m_{}</math> and <math>\displaystyle n_{}</math> are relatively prime positive integers.  Find <math>\displaystyle m+n.</math>
+Point <math>P_{}</math> is located inside traingle <math>ABC</math> so that angles <math>PAB, PBC,</math> and <math>PCA</math> are all congruent.  The sides of the triangle have lengths <math>AB=13, BC=14,</math> and <math>CA=15,</math> and the tangent of angle <math>PAB</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers.  Find <math>m+n.</math>
+__TOC__
 == Solution ==
-{{solution}}
+[[Image:1999_AIME-14.png]]
+=== Solution 1 ===
+Drop perpendiculars from <math>P</math> to the three sides of <math>\triangle ABC</math> and let them meet <math>\overline{AB}, \overline{BC},</math> and <math>\overline{CA}</math> at <math>D, E,</math> and <math>F</math> respectively.  Let <math>BE = x, CF = y,</math> and <math>AD = z</math>.  We have that
+<cmath>
+\begin{eqnarray*} DP & = & z\tan \theta \\
+EP & = & x\tan \theta \\
+FP & = & y\tan \theta \end{eqnarray*}
+</cmath>
+We can then use the tool of calculating area in two ways
+<cmath>
+\begin{eqnarray*} [ABC] & = & [PAB] + [PBC] + [PCA] \\
+& = & \frac 12 (13)(z\tan \theta) + \frac 12 (14)(x\tan\theta) + \frac 12 (15)(y\tan\theta) \\
+& = & \frac 12 \tan\theta(13z + 14x + 15y) \end{eqnarray*}
+</cmath>
+On the other hand
+<cmath>
+\begin{eqnarray*} [ABC] & = & \sqrt {s(s - a)(s - b)(s - c)} \\
+& = & \sqrt {21\cdot 6\cdot 7\cdot 8} \\
+& = & 84 \end{eqnarray*}
+</cmath>
+We still need <math>13z + 14x + 15y</math> though.  We have all these right triangles and we haven't even touched Pythagoras.  So we give it a shot
+<cmath>
+\begin{eqnarray} x^2 + x^2\tan^2\theta & = & z^2\tan^2\theta + (13 - z)^2 \\
+z^2 + z^2\tan^2\theta & = & y^2\tan^2\theta + (15 - y)^2 \\
+y^2 + y^2\tan^2\theta & = & x^2\tan^2\theta + (14 - x)^2 \\
+\end{eqnarray*}
+</cmath>
+But then <math>(1) + (2) + (3)</math> gives
+<cmath>
+\begin{eqnarray*} x^2 + y^2 + z^2 & = & (14 - x)^2 + (15 - y)^2 + (13 - z)^2 \\
+\Rightarrow 13z + 14x + 15y & = & 295 \end{eqnarray*}
+</cmath>
+Recall that we found that <math>[ABC] = \frac 12 \tan\theta(13z + 14x + 15y) = 84</math>.  Plugging in <math>13z + 14x + 15y = 295</math> we get <math>\tan \theta = \frac {168}{295}</math> giving us <math>\boxed{463}</math> for an answer.
+=== Solution 2 ===
+Let <math>AB = c, BC = a, AC = b, PA = x, PB = y, PC = z</math>.
+So by the Law of Cosines, we have:
+<math>x^2 = z^2 + b^2 - 2bz\cos{\theta}</math>
+<math>y^2 = x^2 + c^2 - 2cx\cos{\theta}</math>
+<math>z^2 = y^2 + a^2 - 2ay\cos{\theta}</math>
+Adding these equations and rearranging, we have:
+<math>a^2 + b^2 + c^2 = (2bz + 2cx + 2ay)\cos{\theta}</math>                (1)
+Now <math>[CAP] + [ABP] + [BCP] = [ABC] = \sqrt {(21)(8)(7)(6)} = 84</math>, by Heron's Formula.
+Now the area of a triangle, <math>[A] = \frac {mn\sin{\beta}}{2}</math>, where <math>m</math> and <math>n</math> are sides on either side of an angle, <math>\beta</math>. So,
+<math>[CAP] = \frac {bz\sin{\theta}}{2}</math>.
+<math>[ABP] = \frac {cx\sin{\theta}}{2}</math>.
+<math>[BCP] = \frac {ay\sin{\theta}}{2}</math>
+So adding these equations yields:
+<math>[ABC] = 84 = \frac {(bz + cx + ay)\sin{\theta}}{2}</math>
+<math>\Rightarrow 168 = (bz + cx + ay)\sin{\theta}</math>                                        (2)
+Dividing (2) by (1), we have:
+<math>\frac {168}{a^2 + b^2 + c^2} = \frac {(bz + cx + ay)\sin{\theta}}{(2bz + 2cx + 2ay)\cos{\theta}}</math>
+<math>\Rightarrow \tan{\theta} = \frac {336}{a^2 + b^2 + c^2} = \frac {336}{14^2 + 15^2 + 13^2} = \frac {336}{590} = \frac {168}{295}</math>
+So <math>m + n = 168 + 295 = \boxed{463}</math>.
 == See also ==
+*[[Brocard point]]
 {{AIME box|year=1999|num-b=13|num-a=15}}
+[[Category:Intermediate Geometry Problems]]

1999 AIME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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