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Difference between revisions of "2008 AMC 12B Problems/Problem 17"

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C)18
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Let the coordinates of <math>A</math> be <math>(m, m^2)</math> and the coordinates of <math>C</math> be <math>(n, n^2)</math>. Since the line <math>AB</math> is parallel to the <math>x</math>-axis, the coordinates of <math>B</math> must then be <math>(-m, m^2)</math>.
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Then the slope of line <math>AC</math> is <math>\frac{m^2-n^2}{m-n}=\frac{(m+n)(m-n)}{m-n}=m+n</math>.
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The slope of line <math>BC</math> is <math>\frac{m^2-n^2}{-m-n}=-\frac{(m+n)(m-n)}{m+n}=-(m-n)</math>.
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Supposing <math>\angle A=90^\circ</math>, then <math>AC</math> is perpendicular to <math>AB</math> and, it follows, to the <math>x</math>-axis, making <math>AB</math>  a segment of the line x=m. But that would mean that the coordinates of <math>C</math> <math>(m, m^2)</math>, contradicting the given that points <math>A</math> and <math>C</math> are distinct. So <math>\angle A</math> is not <math>90^\circ</math>. By a similar logic, neither is <math>\angle B</math>.
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This means that <math>\angle C=90^\circ</math> and <math>AC</math> is perpendicular to <math>BC</math>. So the slope of <math>BC</math> is the negative reciprocal of the slope of <math>AC</math>, yielding <math>m+n=\frac{1}{m-n}</math> <math>\Rightarrow</math> <math>m^2-n^2=1</math>.
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Because <math>m^2-n^2</math> is the length of the altitude of triangle <math>ABC</math> from <math>AB</math>, and <math>2m</math> is the length of <math>AB</math>, the area of <math>\triangle ABC=m(m^2-n^2)=2008</math>. Since <math>m^2-n^2=1</math>, <math>m=2008</math>.
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Substituting, <math>2008^2-n^2=1</math> <math>\Rightarrow</math> <math>n^2=2008^2-1=4032063</math>, whose digits sum to <math>18 \Rightarrow \textbf{(C)}</math>.

Revision as of 01:34, 23 December 2009

Let the coordinates of $A$ be $(m, m^2)$ and the coordinates of $C$ be $(n, n^2)$. Since the line $AB$ is parallel to the $x$-axis, the coordinates of $B$ must then be $(-m, m^2)$. Then the slope of line $AC$ is $\frac{m^2-n^2}{m-n}=\frac{(m+n)(m-n)}{m-n}=m+n$. The slope of line $BC$ is $\frac{m^2-n^2}{-m-n}=-\frac{(m+n)(m-n)}{m+n}=-(m-n)$.

Supposing $\angle A=90^\circ$, then $AC$ is perpendicular to $AB$ and, it follows, to the $x$-axis, making $AB$ a segment of the line x=m. But that would mean that the coordinates of $C$ $(m, m^2)$, contradicting the given that points $A$ and $C$ are distinct. So $\angle A$ is not $90^\circ$. By a similar logic, neither is $\angle B$.

This means that $\angle C=90^\circ$ and $AC$ is perpendicular to $BC$. So the slope of $BC$ is the negative reciprocal of the slope of $AC$, yielding $m+n=\frac{1}{m-n}$ $\Rightarrow$ $m^2-n^2=1$.

Because $m^2-n^2$ is the length of the altitude of triangle $ABC$ from $AB$, and $2m$ is the length of $AB$, the area of $\triangle ABC=m(m^2-n^2)=2008$. Since $m^2-n^2=1$, $m=2008$. Substituting, $2008^2-n^2=1$ $\Rightarrow$ $n^2=2008^2-1=4032063$, whose digits sum to $18 \Rightarrow \textbf{(C)}$.

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