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Difference between revisions of "2013 AMC 10B Problems/Problem 23"

(Created page with "In triangle <math>ABC</math>, <math>AB = 13</math>, <math>BC = 14</math>, and <math>CA = 15</math>. Distinct points <math>D</math>, <math>E</math>, and <math>F</math> lie on segm...")
 
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==Problem==
 
In triangle <math>ABC</math>, <math>AB = 13</math>, <math>BC = 14</math>, and <math>CA = 15</math>. Distinct points <math>D</math>, <math>E</math>, and <math>F</math> lie on segments <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{DE}</math>, respectively, such that <math>\overline{AD} \perp \overline{BC}</math>, <math>\overline{DE} \perp \overline{AC}</math>, and <math>\overline{AF} \perp \overline{BF}</math>. The length of segment <math>\overline{DF}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
 
In triangle <math>ABC</math>, <math>AB = 13</math>, <math>BC = 14</math>, and <math>CA = 15</math>. Distinct points <math>D</math>, <math>E</math>, and <math>F</math> lie on segments <math>\overline{BC}</math>, <math>\overline{CA}</math>, and <math>\overline{DE}</math>, respectively, such that <math>\overline{AD} \perp \overline{BC}</math>, <math>\overline{DE} \perp \overline{AC}</math>, and <math>\overline{AF} \perp \overline{BF}</math>. The length of segment <math>\overline{DF}</math> can be written as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
  

Revision as of 16:49, 21 February 2013

Problem

In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD} \perp \overline{BC}$, $\overline{DE} \perp \overline{AC}$, and $\overline{AF} \perp \overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?


$\textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 30$

Solution

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