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Difference between revisions of "2022 SSMO Relay Round 1 Problems/Problem 2"

(Created page with "==Problem== Let <math>T=</math> TNYWR. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math>...")
 
 
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==Problem==
 
==Problem==
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Let <math>T=</math> TNYWR. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math> respectively. Suppose that <math>A</math> is a point on <math>\ell,</math> such that the shortest distance from <math>A</math> to <math>m</math> is <math>T</math>. Given that <math>O</math> is a point on <math>m</math> such that <math>\overline{AO}\perp m,</math> and <math>P</math> is a point on <math>\ell</math> such that <math>PO\perp \ell</math>, find <math>PO^2.</math>
+
Let <math>T=TNYWR</math>. Now, let <math>\ell</math> and <math>m</math> have equations <math>y=(2+\sqrt{3})x+16</math> and <math>y=\frac{x\sqrt{3}}{3}+20,</math> respectively. Suppose that <math>A</math> is a point on <math>\ell,</math> such that the shortest distance from <math>A</math> to <math>m</math> is <math>T</math>. Given that <math>O</math> is a point on <math>m</math> such that <math>\overline{AO}\perp m,</math> and <math>P</math> is a point on <math>\ell</math> such that <math>PO\perp \ell</math>, find <math>PO^2.</math>
  
 
==Solution==
 
==Solution==

Latest revision as of 19:22, 2 May 2025

Problem

Let $T=TNYWR$. Now, let $\ell$ and $m$ have equations $y=(2+\sqrt{3})x+16$ and $y=\frac{x\sqrt{3}}{3}+20,$ respectively. Suppose that $A$ is a point on $\ell,$ such that the shortest distance from $A$ to $m$ is $T$. Given that $O$ is a point on $m$ such that $\overline{AO}\perp m,$ and $P$ is a point on $\ell$ such that $PO\perp \ell$, find $PO^2.$

Solution

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