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Difference between revisions of "1986 AIME Problems/Problem 9"

m (→‎Solution 1)
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== Problem ==
 
== Problem ==
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In <math>\displaystyle \triangle ABC</math>, <math>\displaystyle AB= 425</math>, <math>\displaystyle BC=450</math>, and <math>\displaystyle AC=510</math>. An interior [[point]] <math>\displaystyle P</math> is then drawn, and [[segment]]s are drawn through <math>\displaystyle P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>\displaystyle d</math>, find <math>\displaystyle d</math>.
+
In <math>\triangle ABC</math>, <math>AB= 425</math>, <math>BC=450</math>, and <math>AC=510</math>. An interior [[point]] <math>P</math> is then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>d</math>, find <math>d</math>.
  
 
__TOC__
 
__TOC__
 
== Solution ==
 
== Solution ==
 
=== Solution 1 ===
 
=== Solution 1 ===
−
[[Image:1986_AIME-9.png]]
+
[[Image:1986_AIME-9.png|center]]
  
 
Let the points at which the segments hit the triangle be called <math>D, D', E, E', F, F'</math> as shown above. All three smaller triangles and the larger triangle are [[similar triangles|similar]] (<math>\triangle ABC \sim \triangle DPD' \sim \triangle PEE' \sim \triangle F'PF</math>). This is easy to find using repeated [[alternate interior angles]]. The remaining three sections are [[parallelogram]]s, which is also simple to see by the parallel lines.  
 
Let the points at which the segments hit the triangle be called <math>D, D', E, E', F, F'</math> as shown above. All three smaller triangles and the larger triangle are [[similar triangles|similar]] (<math>\triangle ABC \sim \triangle DPD' \sim \triangle PEE' \sim \triangle F'PF</math>). This is easy to find using repeated [[alternate interior angles]]. The remaining three sections are [[parallelogram]]s, which is also simple to see by the parallel lines.  
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Since <math>\triangle DPD' \sim \triangle ABC</math>, we have the [[proportion]]:
 
Since <math>\triangle DPD' \sim \triangle ABC</math>, we have the [[proportion]]:
  
−
<div style="text-align:center;"><math>\frac{425-d}{425} = \frac{PD}{510}</math><br /><math>PD = 510 - \frac{510}{425}d = 510 - \frac{6}{5}d</math></div>
+
<cmath>\frac{425-d}{425} = \frac{PD}{510} \Longrightarrow PD = 510 - \frac{510}{425}d = 510 - \frac{6}{5}d</cmath>
  
−
Doing the same with <math>\triangle PEE'</math>, we find that <math>PE' =510 - \frac{17}{15}d</math>. Now, <math>d = PD + PE' = 510 - \frac{6}{5}d + 510 - \frac{17}{15}d \Longrightarrow d\left(\frac{50}{15}\right) = 1020 \Longrightarrow d = 306</math>.
+
Doing the same with <math>\triangle PEE'</math>, we find that <math>PE' =510 - \frac{17}{15}d</math>. Now, <math>d = PD + PE' = 510 - \frac{6}{5}d + 510 - \frac{17}{15}d \Longrightarrow d\left(\frac{50}{15}\right) = 1020 \Longrightarrow d = \boxed{306}</math>.
  
 
=== Solution 2 ===
 
=== Solution 2 ===
 
Define the points the same as above.
 
Define the points the same as above.
  
−
Let <math>[CE'PF] = a</math>, <math>[E'EP] = b</math>, <math>\displaystyle [BEPD'] = c \displaystyle</math>, <math>[D'PD] = d</math>, <math>\displaystyle [DAF'P] = e</math> and <math>[F'D'P] = f</math>
+
Let <math>[CE'PF] = a</math>, <math>[E'EP] = b</math>, <math>[BEPD'] = c</math>, <math>[D'PD] = d</math>, <math>[DAF'P] = e</math> and <math>[F'D'P] = f</math>
  
 
Key theorem: the ratio of the areas of 2 similar triangles is the ratio of a pair of corresponding sides squared.
 
Key theorem: the ratio of the areas of 2 similar triangles is the ratio of a pair of corresponding sides squared.
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Let the length of the segment be <math>x</math> and the area of the triangle be <math>A</math>, using the theorem, we get:
 
Let the length of the segment be <math>x</math> and the area of the triangle be <math>A</math>, using the theorem, we get:
  
−
<math>\frac {c + e + d}{A} = \left(\frac {x}{BC}\right)^2</math>, <math>\displaystyle \frac {b + c + d}{A} \displaystyle = \left(\frac {x}{AC}\right)^2</math>, <math>\frac {a + b + f}{A} = \left(\frac {x}{AB}\right)^2</math>
+
<math>\frac {c + e + d}{A} = \left(\frac {x}{BC}\right)^2</math>, <math>\frac {b + c + d}{A}= \left(\frac {x}{AC}\right)^2</math>, <math>\frac {a + b + f}{A} = \left(\frac {x}{AB}\right)^2</math>
 
adding all these together and using <math>a + b + c + d + e + f = A</math> we get
 
adding all these together and using <math>a + b + c + d + e + f = A</math> we get
 
<math>\frac {f + d + b}{A} + 1 = x^2*\left(\frac {1}{BC^2} + \frac {1}{AC^2} + \frac {1}{AB^2}\right)</math>
 
<math>\frac {f + d + b}{A} + 1 = x^2*\left(\frac {1}{BC^2} + \frac {1}{AC^2} + \frac {1}{AB^2}\right)</math>
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Now we have the side length [[ratio]], so we have the area ratio
 
Now we have the side length [[ratio]], so we have the area ratio
−
<math>\displaystyle \frac {f}{A} = \left(\frac {AC - x}{AC}\right)^2 = \left(1 - \frac {x}{AC}\right)^2</math>, by symmetry, we have
+
<math>\frac {f}{A} = \left(\frac {AC - x}{AC}\right)^2 = \left(1 - \frac {x}{AC}\right)^2</math>, by symmetry, we have
−
<math>\displaystyle \frac {d}{A} = \left(1 - \frac {x}{AB}\right)^2</math> and <math>\frac {b}{A} = \left(1 - \frac {x}{BC}\right)^2</math>
+
<math>\frac {d}{A} = \left(1 - \frac {x}{AB}\right)^2</math> and <math>\frac {b}{A} = \left(1 - \frac {x}{BC}\right)^2</math>
  
 
Substituting these into our initial equation, we have
 
Substituting these into our initial equation, we have
 
<math>1 + \sum_{cyc}\left(1 - \frac {x}{AB}\right) - \frac {x^2}{AB^2} = 0</math>
 
<math>1 + \sum_{cyc}\left(1 - \frac {x}{AB}\right) - \frac {x^2}{AB^2} = 0</math>
 
<math>1 + \sum_{cyc}1 - 2*\frac {x}{AB} = 0</math>
 
<math>1 + \sum_{cyc}1 - 2*\frac {x}{AB} = 0</math>
−
<math>\displaystyle \frac {2}{\frac {1}{AB} + \frac {1}{BC} + \frac {1}{CA}} = x</math>
+
<math>\frac {2}{\frac {1}{AB} + \frac {1}{BC} + \frac {1}{CA}} = x</math>
−
answer follows after some hideous computation
+
answer follows after some hideous computation.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=1986|num-b=8|num-a=10}}
 
{{AIME box|year=1986|num-b=8|num-a=10}}
−
* [[AIME Problems and Solutions]]
 
−
* [[American Invitational Mathematics Examination]]
 
−
* [[Mathematics competition resources]]
 
  
 +
[[Category:Asymptote needed]]
 
[[Category:Intermediate Geometry Problems]]
 
[[Category:Intermediate Geometry Problems]]

Revision as of 19:34, 9 April 2008

Problem

In $\triangle ABC$, $AB= 425$, $BC=450$, and $AC=510$. An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these three segments are of an equal length $d$, find $d$.

Solution

Solution 1

1986 AIME-9.png

Let the points at which the segments hit the triangle be called $D, D', E, E', F, F'$ as shown above. All three smaller triangles and the larger triangle are similar ($\triangle ABC \sim \triangle DPD' \sim \triangle PEE' \sim \triangle F'PF$). This is easy to find using repeated alternate interior angles. The remaining three sections are parallelograms, which is also simple to see by the parallel lines.

Since $PDAF'$ is a parallelogram, we find $PF' = AD$, and similarly $PE = BD'$. So $d = PF' + PE = AD + BD' = 425 - DD'$. Thus $DD' = 425 - d$. By the same logic, $EE' = 450 - d$.

Since $\triangle DPD' \sim \triangle ABC$, we have the proportion:

\[\frac{425-d}{425} = \frac{PD}{510} \Longrightarrow PD = 510 - \frac{510}{425}d = 510 - \frac{6}{5}d\]

Doing the same with $\triangle PEE'$, we find that $PE' =510 - \frac{17}{15}d$. Now, $d = PD + PE' = 510 - \frac{6}{5}d + 510 - \frac{17}{15}d \Longrightarrow d\left(\frac{50}{15}\right) = 1020 \Longrightarrow d = \boxed{306}$.

Solution 2

Define the points the same as above.

Let $[CE'PF] = a$, $[E'EP] = b$, $[BEPD'] = c$, $[D'PD] = d$, $[DAF'P] = e$ and $[F'D'P] = f$

Key theorem: the ratio of the areas of 2 similar triangles is the ratio of a pair of corresponding sides squared.

Let the length of the segment be $x$ and the area of the triangle be $A$, using the theorem, we get:

$\frac {c + e + d}{A} = \left(\frac {x}{BC}\right)^2$, $\frac {b + c + d}{A}= \left(\frac {x}{AC}\right)^2$, $\frac {a + b + f}{A} = \left(\frac {x}{AB}\right)^2$ adding all these together and using $a + b + c + d + e + f = A$ we get $\frac {f + d + b}{A} + 1 = x^2*\left(\frac {1}{BC^2} + \frac {1}{AC^2} + \frac {1}{AB^2}\right)$

Using corresponding angles from parallel lines, it is easy to show that $\triangle ABC \sim \triangle F'PF$, since $ADPF'$ and $CFPE'$ are parallelograms, it is easy to show that $FF' = AC - x$

Now we have the side length ratio, so we have the area ratio $\frac {f}{A} = \left(\frac {AC - x}{AC}\right)^2 = \left(1 - \frac {x}{AC}\right)^2$, by symmetry, we have $\frac {d}{A} = \left(1 - \frac {x}{AB}\right)^2$ and $\frac {b}{A} = \left(1 - \frac {x}{BC}\right)^2$

Substituting these into our initial equation, we have $1 + \sum_{cyc}\left(1 - \frac {x}{AB}\right) - \frac {x^2}{AB^2} = 0$ $1 + \sum_{cyc}1 - 2*\frac {x}{AB} = 0$ $\frac {2}{\frac {1}{AB} + \frac {1}{BC} + \frac {1}{CA}} = x$ answer follows after some hideous computation.

See also

1986 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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All AIME Problems and Solutions
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