Difference between revisions of "1983 AHSME Problems/Problem 29"

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==Problem==
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A point <math>P</math> lies in the same plane as a given square of side <math>1</math>. Let the vertices of the square,
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taken counterclockwise, be <math>A, B, C</math> and <math>D</math>. Also, let the distances from <math>P</math> to <math>A, B</math> and <math>C</math>, respectively, be <math>u, v</math> and <math>w</math>.
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What is the greatest distance that <math>P</math> can be from <math>D</math> if <math>u^2 + v^2 = w^2</math>?
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<math>\textbf{(A)}\ 1 + \sqrt{2}\qquad
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\textbf{(B)}\ 2\sqrt{2}\qquad
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\textbf{(C)}\ 2 + \sqrt{2}\qquad
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\textbf{(D)}\ 3\sqrt{2}\qquad
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\textbf{(E)}\ 3+\sqrt{2}    </math>
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==Solution==
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Place the square on the coordinate plane with <math>A</math> as the origin. (This means that <math>B=(1,0), C=(1,1),</math> and <math>D=(0,1).</math>We are given that <math>PA^2+PB^2=PC^2,</math> so  
 
Place the square on the coordinate plane with <math>A</math> as the origin. (This means that <math>B=(1,0), C=(1,1),</math> and <math>D=(0,1).</math>We are given that <math>PA^2+PB^2=PC^2,</math> so  
  
\begin{align*}&(x^2+y^2)+((x-1)^2+y^2)=(x-1)^2+(y-1)^2\\ &2x^2+2y^2-2x+1=x^2+y^2-2x-2y+2\\ &x^2+y^2=-2y+1\\ &x^2+y^2+2y-1=0\\ &x^2+(y+1)^2=2\end{align*}
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<cmath>\begin{align*}&(x^2+y^2)+((x-1)^2+y^2)=(x-1)^2+(y-1)^2\\ &2x^2+2y^2-2x+1=x^2+y^2-2x-2y+2\\ &x^2+y^2=-2y+1\\ &x^2+y^2+2y-1=0\\ &x^2+(y+1)^2=2\end{align*}</cmath>
  
 
Thus, we see that <math>P</math> is on a circle centered at <math>(0,-1)</math> with radius <math>\sqrt{2}.</math> The farthest point from <math>D</math> on this circle is at the bottom of the circle, at <math>(0, -1-\sqrt{2}),</math> so <math>PD</math> is <math>\boxed{2+\sqrt{2}}.</math>
 
Thus, we see that <math>P</math> is on a circle centered at <math>(0,-1)</math> with radius <math>\sqrt{2}.</math> The farthest point from <math>D</math> on this circle is at the bottom of the circle, at <math>(0, -1-\sqrt{2}),</math> so <math>PD</math> is <math>\boxed{2+\sqrt{2}}.</math>

Revision as of 18:53, 27 January 2019

Problem

A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 = w^2$?

$\textbf{(A)}\ 1 + \sqrt{2}\qquad \textbf{(B)}\ 2\sqrt{2}\qquad \textbf{(C)}\ 2 + \sqrt{2}\qquad \textbf{(D)}\ 3\sqrt{2}\qquad \textbf{(E)}\ 3+\sqrt{2}$

Solution

Place the square on the coordinate plane with $A$ as the origin. (This means that $B=(1,0), C=(1,1),$ and $D=(0,1).$We are given that $PA^2+PB^2=PC^2,$ so

\begin{align*}&(x^2+y^2)+((x-1)^2+y^2)=(x-1)^2+(y-1)^2\\ &2x^2+2y^2-2x+1=x^2+y^2-2x-2y+2\\ &x^2+y^2=-2y+1\\ &x^2+y^2+2y-1=0\\ &x^2+(y+1)^2=2\end{align*}

Thus, we see that $P$ is on a circle centered at $(0,-1)$ with radius $\sqrt{2}.$ The farthest point from $D$ on this circle is at the bottom of the circle, at $(0, -1-\sqrt{2}),$ so $PD$ is $\boxed{2+\sqrt{2}}.$

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