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Difference between revisions of "2001 AIME II Problems/Problem 7"

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(solutions/asys; solution (2) by Altheman)
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== Problem ==
 
== Problem ==
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Let <math>\triangle{PQR}</math> be a right triangle with <math>PQ = 90</math>, <math>PR = 120</math>, and <math>QR = 150</math>. Let <math>C_{1}</math> be the inscribed circle. Construct <math>\overline{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> is perpendicular to <math>\overline{PR}</math> and tangent to <math>C_{1}</math>. Construct <math>\overline{UV}</math> with <math>U</math> on <math>\overline{PQ}</math> and <math>V</math> on <math>\overline{QR}</math> such that <math>\overline{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <math>\triangle{RST}</math> and <math>C_{3}</math> the inscribed circle of <math>\triangle{QUV}</math>. The distance between the centers of <math>C_{2}</math> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>?
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Let <math>\triangle{PQR}</math> be a [[right triangle]] with <math>PQ = 90</math>, <math>PR = 120</math>, and <math>QR = 150</math>. Let <math>C_{1}</math> be the [[incircle|inscribed circle]]. Construct <math>\overline{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{QR}</math>, such that <math>\overline{ST}</math> is [[perpendicular]] to <math>\overline{PR}</math> and tangent to <math>C_{1}</math>. Construct <math>\overline{UV}</math> with <math>U</math> on <math>\overline{PQ}</math> and <math>V</math> on <math>\overline{QR}</math> such that <math>\overline{UV}</math> is perpendicular to <math>\overline{PQ}</math> and tangent to <math>C_{1}</math>. Let <math>C_{2}</math> be the inscribed circle of <math>\triangle{RST}</math> and <math>C_{3}</math> the inscribed circle of <math>\triangle{QUV}</math>. The distance between the centers of <math>C_{2}</math> and <math>C_{3}</math> can be written as <math>\sqrt {10n}</math>. What is <math>n</math>?
  
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__TOC__
 
== Solution ==
 
== Solution ==
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{{solution}}
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=== Solution 1 (analytic) ===
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<center><asy>
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pointpen = black; pathpen = black + linewidth(0.7);
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pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10);
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D(MP("P",P)--MP("Q",Q)--MP("R",R,W)--cycle); D(MP("S",S,W) -- MP("T",T,NE)); D(MP("U",U) -- MP("V",V,NE)); D(O2 -- O3, rgb(0.2,0.5,0.2)+ linewidth(0.7) + linetype("4 4"));
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D(CR(D(O1), 30)); D(CR(D(O2), 15)); D(CR(D(O3), 10));
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</asy></center>
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Let <math>P = (0,0)</math> be at the origin. Using the formula <math>A = rs</math> on <math>\triangle PQR</math>, where <math>r_{1}</math> is the [[inradius]] (similarly define <math>r_2, r_3</math> to be the radii of <math>C_2, C_3</math>), <math>s = \frac{PQ + QR + RP}{2} = 180</math> is the [[semiperimeter]], and <math>A = \frac 12 bh = 5400</math> is the area, we find <math>r_{1} = \frac As = 30</math>. Thus <math>ST, UV</math> lie respectively on the lines <math>y = 60, x = 60</math>, and so <math>RS = 60, UQ = 30</math>. 
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Note that <math>\triangle PQR \sim \triangle STR \sim \triangle UQV</math>. Since the ratio of corresponding lengths of similar figures are the same, we have
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<cmath>\frac{r_{1}}{PR} = \frac{r_{2}}{RS} \Longrightarrow r_{2} = 15\ \text{and} \ \frac{r_{1}}{PQ} = \frac{r_{3}}{UQ} \Longrightarrow r_{3} = 10.</cmath>
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Let the centers of <math>\odot C_2, C_3</math> be <math>O_2 = (0 + r_{2}, 60 + r_{2}) = (15, 75), O_3 = (60 + r_{3}, 0 + r_{3}) = (70,10)</math>, respectively; then by the [[distance formula]] we have <math>O_2O_3 = \sqrt{55^2 + 65^2} = \sqrt{10 \cdot 725}</math>. Therefore, the answer is <math>n = \boxed{725}</math>.
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=== Solution 2 (synthetic) ===
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<center><asy>
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pointpen = black; pathpen = black + linewidth(0.7);
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pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10);
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D(MP("P",P)--MP("Q",Q)--MP("R",R,W)--cycle); D(MP("S",S,W) -- MP("T",T,NE)); D(MP("U",U) -- MP("V",V,NE)); D(O2 -- O3, rgb(0.2,0.5,0.2)+ linewidth(0.7) + linetype("4 4"));
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D(CR(D(O1), 30)); D(CR(D(O2), 15)); D(CR(D(O3), 10));
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pair A2 = IP(incircle(R,S,T), Q--R), A3 = IP(incircle(Q,U,V), Q--R);
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D(D(MP("A_2",A2,NE)) -- O2, linetype("4 4")+linewidth(0.6)); D(D(MP("A_3",A3,NE)) -- O3 -- foot(O3, A2, O2), linetype("4 4")+linewidth(0.6));
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</asy></center>
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We compute <math>r_1 = 30, r_2 = 15, r_3 = 10</math> as above. Let <math>A_1, A_2, A_3</math> respectively the points of tangency of <math>C_1, C_2, C_3</math> with <math>QR</math>.
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By the [[Two Tangent Theorem]], we find that <math>A_{1}Q = 60</math>, <math>A_{1}R = 90</math>. Using the similar triangles, <math>RA_{2} = 45</math>, <math>QA_{3} = 20</math>, so <math>A_{2}A_{3} = QR - RA_2 - QA_3 = 85</math>. Thus <math>(O_{2}O_{3})^{2} = (15 - 10)^{2} + (85)^{2} = 7250\implies n=\boxed{725}</math>.
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2001|n=II|num-b=6|num-a=8}}
 
{{AIME box|year=2001|n=II|num-b=6|num-a=8}}
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[[Category:Intermediate Geometry Problems]]

Revision as of 13:44, 26 July 2008

Problem

Let $\triangle{PQR}$ be a right triangle with $PQ = 90$, $PR = 120$, and $QR = 150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt {10n}$. What is $n$?

Solution

Solution 1 (analytic)

[asy] pointpen = black; pathpen = black + linewidth(0.7); pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10); D(MP("P",P)--MP("Q",Q)--MP("R",R,W)--cycle); D(MP("S",S,W) -- MP("T",T,NE)); D(MP("U",U) -- MP("V",V,NE)); D(O2 -- O3, rgb(0.2,0.5,0.2)+ linewidth(0.7) + linetype("4 4")); D(CR(D(O1), 30)); D(CR(D(O2), 15)); D(CR(D(O3), 10)); [/asy]

Let $P = (0,0)$ be at the origin. Using the formula $A = rs$ on $\triangle PQR$, where $r_{1}$ is the inradius (similarly define $r_2, r_3$ to be the radii of $C_2, C_3$), $s = \frac{PQ + QR + RP}{2} = 180$ is the semiperimeter, and $A = \frac 12 bh = 5400$ is the area, we find $r_{1} = \frac As = 30$. Thus $ST, UV$ lie respectively on the lines $y = 60, x = 60$, and so $RS = 60, UQ = 30$.

Note that $\triangle PQR \sim \triangle STR \sim \triangle UQV$. Since the ratio of corresponding lengths of similar figures are the same, we have

\[\frac{r_{1}}{PR} = \frac{r_{2}}{RS} \Longrightarrow r_{2} = 15\ \text{and} \ \frac{r_{1}}{PQ} = \frac{r_{3}}{UQ} \Longrightarrow r_{3} = 10.\]

Let the centers of $\odot C_2, C_3$ be $O_2 = (0 + r_{2}, 60 + r_{2}) = (15, 75), O_3 = (60 + r_{3}, 0 + r_{3}) = (70,10)$, respectively; then by the distance formula we have $O_2O_3 = \sqrt{55^2 + 65^2} = \sqrt{10 \cdot 725}$. Therefore, the answer is $n = \boxed{725}$.

Solution 2 (synthetic)

[asy] pointpen = black; pathpen = black + linewidth(0.7); pair P = (0,0), Q = (90, 0), R = (0, 120), S=(0, 60), T=(45, 60), U = (60,0), V=(60, 40), O1 = (30,30), O2 = (15, 75), O3 = (70, 10); D(MP("P",P)--MP("Q",Q)--MP("R",R,W)--cycle); D(MP("S",S,W) -- MP("T",T,NE)); D(MP("U",U) -- MP("V",V,NE)); D(O2 -- O3, rgb(0.2,0.5,0.2)+ linewidth(0.7) + linetype("4 4")); D(CR(D(O1), 30)); D(CR(D(O2), 15)); D(CR(D(O3), 10)); pair A2 = IP(incircle(R,S,T), Q--R), A3 = IP(incircle(Q,U,V), Q--R); D(D(MP("A_2",A2,NE)) -- O2, linetype("4 4")+linewidth(0.6)); D(D(MP("A_3",A3,NE)) -- O3 -- foot(O3, A2, O2), linetype("4 4")+linewidth(0.6)); [/asy]

We compute $r_1 = 30, r_2 = 15, r_3 = 10$ as above. Let $A_1, A_2, A_3$ respectively the points of tangency of $C_1, C_2, C_3$ with $QR$.

By the Two Tangent Theorem, we find that $A_{1}Q = 60$, $A_{1}R = 90$. Using the similar triangles, $RA_{2} = 45$, $QA_{3} = 20$, so $A_{2}A_{3} = QR - RA_2 - QA_3 = 85$. Thus $(O_{2}O_{3})^{2} = (15 - 10)^{2} + (85)^{2} = 7250\implies n=\boxed{725}$.

See also

2001 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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