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Difference between revisions of "1984 AIME Problems/Problem 3"
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== Problem ==
 
== Problem ==
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A point <math>\displaystyle P</math> is chosen in the interior of <math>\displaystyle \triangle ABC</math> such that when lines are drawn through <math>\displaystyle P</math> parallel to the sides of <math>\displaystyle \triangle ABC</math>, the resulting smaller triangles <math>\displaystyle t_{1}</math>, <math>\displaystyle t_{2}</math>, and <math>\displaystyle t_{3}</math> in the figure, have areas <math>\displaystyle 4</math>, <math>\displaystyle 9</math>, and <math>\displaystyle 49</math>, respectively. Find the area of <math>\displaystyle \triangle ABC</math>.
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A [[point]] <math>\displaystyle P</math> is chosen in the interior of <math>\displaystyle \triangle ABC</math> such that when [[line]]s are drawn through <math>\displaystyle P</math> [[parallel]] to the sides of <math>\displaystyle \triangle ABC</math>, the resulting smaller [[triangle]]s <math>\displaystyle t_{1}</math>, <math>\displaystyle t_{2}</math>, and <math>\displaystyle t_{3}</math> in the figure, have [[area]]s <math>\displaystyle 4</math>, <math>\displaystyle 9</math>, and <math>\displaystyle 49</math>, respectively. Find the area of <math>\displaystyle \triangle ABC</math>.
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== Solution ==
 
== Solution ==
 
{{solution}}
 
{{solution}}
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* [[1984 AIME Problems/Problem 4 | Next problem]]
 
* [[1984 AIME Problems/Problem 4 | Next problem]]
 
* [[1984 AIME Problems]]
 
* [[1984 AIME Problems]]
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[[Category:Intermediate Geometry Problems]]

Revision as of 11:13, 24 January 2007

Problem

A point $\displaystyle P$ is chosen in the interior of $\displaystyle \triangle ABC$ such that when lines are drawn through $\displaystyle P$ parallel to the sides of $\displaystyle \triangle ABC$, the resulting smaller triangles $\displaystyle t_{1}$, $\displaystyle t_{2}$, and $\displaystyle t_{3}$ in the figure, have areas $\displaystyle 4$, $\displaystyle 9$, and $\displaystyle 49$, respectively. Find the area of $\displaystyle \triangle ABC$.

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Solution

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See also

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