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Difference between revisions of "2025 AIME II Problems/Problem 6"

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</asy>
  
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== Solution 1 (thorough) ==
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== Solution 1 (Thorough) ==
 
Let <math>GH=2x</math> and <math>GF=2y</math>. Notice that since <math>\overline{BC}</math> is perpendicular to <math>\overline{GH}</math> (can be proven using basic angle chasing) and <math>\overline{BC}</math> is an extension of a diameter of <math>\omega_1</math>, then <math>\overline{CB}</math> is the perpendicular bisector of <math>\overline{GH}</math>. Similarly, since <math>\overline{AD}</math> is perpendicular to <math>\overline{GF}</math> (also provable using basic angle chasing) and <math>\overline{AD}</math> is part of a diameter of <math>\omega_1</math>, then <math>\overline{AD}</math> is the perpendicular bisector of <math>\overline{GF}</math>.
 
Let <math>GH=2x</math> and <math>GF=2y</math>. Notice that since <math>\overline{BC}</math> is perpendicular to <math>\overline{GH}</math> (can be proven using basic angle chasing) and <math>\overline{BC}</math> is an extension of a diameter of <math>\omega_1</math>, then <math>\overline{CB}</math> is the perpendicular bisector of <math>\overline{GH}</math>. Similarly, since <math>\overline{AD}</math> is perpendicular to <math>\overline{GF}</math> (also provable using basic angle chasing) and <math>\overline{AD}</math> is part of a diameter of <math>\omega_1</math>, then <math>\overline{AD}</math> is the perpendicular bisector of <math>\overline{GF}</math>.
  
 
From the [[Pythagorean Theorem]] on <math>\triangle GFH</math>, we have <math>(2x)^2+(2y)^2=12^2</math>, so <math>x^2+y^2=36</math>. To find our second equation for our system, we utilize the triangles given.
 
From the [[Pythagorean Theorem]] on <math>\triangle GFH</math>, we have <math>(2x)^2+(2y)^2=12^2</math>, so <math>x^2+y^2=36</math>. To find our second equation for our system, we utilize the triangles given.
  
−
Let <math>I=\overline{GH}\cap\overline{CB}</math>. Then we know that <math>GFBI</math> is also a rectangle since all of its angles can be shown to be right using basic angle chasing, so <math>FG=IB</math>. We also know that <math>CI+IB=2\cdot 15=30</math>. <math>IA=y</math> and <math>AB=6</math>, so <math>CI=30-y-6=24-y</math>. Notice that <math>CI</math> is a height of <math>\triangle CHG</math>, so its area is <math>\frac{1}{2}(2x)(30-2y)=x(24-y)</math>.
+
Let <math>I=\overline{GH}\cap\overline{CB}</math>. Then we know that <math>GFBI</math> is also a rectangle since all of its angles can be shown to be right using basic angle chasing, so <math>FG=IB</math>. We also know that <math>CI+IB=2\cdot 15=30</math>. <math>IA=y</math> and <math>AB=6</math>, so <math>CI=30-y-6=24-y</math>. Notice that <math>CI</math> is a height of <math>\triangle CHG</math>, so its area is <math>\frac{1}{2}(2x)(24-y)=x(24-y)</math>.
  
 
Next, extend <math>\overline{DA}</math> past <math>A</math> to intersect <math>\omega_2</math> again at <math>D'</math>. Since <math>\overline{BC}</math> is given to be a diameter of <math>\omega_2</math> and <math>\overline{BC}\perp\overline{AD}</math>, then <math>\overline{BC}</math> is the perpendicular bisector of <math>\overline{DD'}</math>; thus <math>DA=D'A</math>. By Power of a Point, we know that <math>CA\cdot AB=DA\cdot AD'</math>. <math>CA=30-6=24</math> and <math>AB=6</math>, so <math>DA\cdot AD'=(DA)^2=24\cdot6=144</math> and <math>DA=D'A=12</math>.
 
Next, extend <math>\overline{DA}</math> past <math>A</math> to intersect <math>\omega_2</math> again at <math>D'</math>. Since <math>\overline{BC}</math> is given to be a diameter of <math>\omega_2</math> and <math>\overline{BC}\perp\overline{AD}</math>, then <math>\overline{BC}</math> is the perpendicular bisector of <math>\overline{DD'}</math>; thus <math>DA=D'A</math>. By Power of a Point, we know that <math>CA\cdot AB=DA\cdot AD'</math>. <math>CA=30-6=24</math> and <math>AB=6</math>, so <math>DA\cdot AD'=(DA)^2=24\cdot6=144</math> and <math>DA=D'A=12</math>.
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We are given that <math>[DGF]=[CHG]</math> (<math>[ABC]</math> denotes the area of figure <math>ABC</math>). As a result, <math>x(24-y)=y(12-x)</math>. This can be simplified to <math>y=2x</math>. Substituting this into the Pythagorean equation yields <math>5x^2=36</math> and <math>x=\frac{6}{\sqrt{5}}</math>. Then <math>y=\frac{12}{\sqrt{5}}</math>.
 
We are given that <math>[DGF]=[CHG]</math> (<math>[ABC]</math> denotes the area of figure <math>ABC</math>). As a result, <math>x(24-y)=y(12-x)</math>. This can be simplified to <math>y=2x</math>. Substituting this into the Pythagorean equation yields <math>5x^2=36</math> and <math>x=\frac{6}{\sqrt{5}}</math>. Then <math>y=\frac{12}{\sqrt{5}}</math>.
  
−
<math>[EFGH]=2x\cdot2y=2\cdot\frac{6}{\sqrt{5}}\cdot2\cdot\frac{12}{\sqrt{5}}=\frac{288}{5}</math>, so the answer is <math>288+5=\boxed{293}</math>. ~eevee9406
+
<math>[EFGH]=2x\cdot2y=2\cdot\frac{6}{\sqrt{5}}\cdot2\cdot\frac{12}{\sqrt{5}}=\frac{288}{5}</math>, so the answer is <math>288+5=\boxed{293}</math>.  
  
−
== Solution 2 (faster) ==
+
~ [https://artofproblemsolving.com/wiki/index.php/User:Eevee9406 eevee9406]
  
−
Denote the intersection of <math>BC</math> and <math>w_1</math> as <math>P</math>, the intersection of <math>BC</math> and <math>GH</math> be <math>Q</math>, and the center of <math>w_2</math> to be <math>O</math>. Additionally, let <math>EF = GH = a, FG = EH = b</math>. We have that <math>CP = 18</math> and <math>PQ = \frac{6-b}{2}</math>. Considering right triangle <math>OAD</math>, <math>AD = 12</math>. Letting <math>R</math> be the intersection of <math>AD</math> and <math>FG</math>, <math>DR = 12 - \frac{b}{2}</math>. Using the equivalent area ratios: <cmath> \frac{a(24-\frac{b}{2})}{2} = \frac{(12-\frac{a}{2})b}{2} </cmath>
+
~ Edited by [https://artofproblemsolving.com/wiki/index.php/User:Aoum aoum]
  
−
This equation gives <math>b=2a</math>. Using pythagorean theorem on triangle <math>GHE</math> gives that <math>a^2+b^2 = 144</math>. Plugging the reuslt <math>b=2a</math> into this equation gives that the area of the triangle is <math>\frac{288}{5} \to \boxed{293}</math>
+
== Solution 2 (Faster) ==
  
−
-Vivdax
+
Denote the intersection of <math>BC</math> and <math>w_1</math> as <math>P</math>, the intersection of <math>BC</math> and <math>GH</math> be <math>Q</math>, and the center of <math>w_2</math> to be <math>O</math>. Additionally, let <math>EF = GH = a, FG = EH = b</math>. We have that <math>CP = 18</math> and <math>PQ = 6-\frac{b}{2}</math>. Considering right triangle <math>OAD</math>, <math>AD = 12</math>. Letting <math>R</math> be the intersection of <math>AD</math> and <math>FG</math>, <math>DR = 12 - \frac{b}{2}</math>. Using the equivalent area ratios: <cmath> \frac{a(24-\frac{b}{2})}{2} = \frac{(12-\frac{a}{2})b}{2} </cmath>
 +
 
 +
This equation gives <math>b=2a</math>. Using the [[Pythagorean Theorem]] on triangle <math>GHE</math> gives that <math>a^2+b^2 = 144</math>. Plugging the reuslt <math>b=2a</math> into this equation gives that the area of the triangle is <math>\frac{288}{5} \to \boxed{293}</math>.
 +
 
 +
~ Vivdax
 +
 
 +
~ Edited by [https://artofproblemsolving.com/wiki/index.php/User:Aoum aoum]
 +
 
 +
~ Additional edits by fermat_sLastAMC
 +
 
 +
== Solution 3 (Not Recommended) ==
 +
You can use your ruler to check that <math>GF=2GH.</math> Then, we have <math>GF^2+GH^2=36,</math> and we solve the system of equations to get <math>\frac{288}{5},</math> so the answer is <math>\boxed{293}.</math>
 +
 
 +
Note: This method is not recommended as the diagrams are not necessarily drawn to scale. However, it can be used in emergency situations or to verify the answer.
 +
 
 +
~derekwang2048
 +
 
 +
 
 +
== Solution 4 (Almost no Algebra) ==
 +
 
 +
If we draw segments connecting <math>A</math> to <math>F, G,</math> and <math>H,</math> we can easily verify that all of the right triangles created are congruent. Thus, triangles <math>AGF</math> and <math>AGH</math> have equal areas, which means, by the given conditions, that kites <math>AGCH</math> and <math>AGDF</math> have equal areas. Thus, by the area formula for kites,
 +
 
 +
<cmath>\frac{1}{2}(AD)(GF)=\frac{1}{2}(AC)(BH),</cmath> or <cmath>(AD)(GF)=(24)(GH).</cmath>
 +
 
 +
Also, if we extend <math>AD</math> to the other side of the large circle, the chord length formula gives
 +
 
 +
<cmath>(AD)(AD)=(AC)(AB),</cmath> so <cmath>(AD)(AD)=(24)(6).</cmath>
 +
 
 +
 
 +
Squaring the second equation above gives
 +
<cmath>(AD)(AD)(GF)(GF)=(24)(24)(GH)(GH).</cmath>
 +
Dividing by the fourth equation gives
 +
<cmath>(GF)^2=4(GH)^2.</cmath>
 +
Since we know, by the Pythagorean Theorem, that <math>(GF)^2+(GH)^2=144,</math> we can substitute to determine that <math>5(GH)^2=144.</math>
 +
 
 +
The desired area is <math>(GF)(GH)=2(GH)^2=\frac{2}{5}(144),</math> which is <math>\frac{288}{5},</math> so the answer is <math>\boxed{293}.</math>
  
 
==See also==
 
==See also==

Latest revision as of 11:32, 16 February 2025

Problem

Circle $\omega_1$ with radius $6$ centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius $15$. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and ${\overline{BC} \perp \overline{AD}}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\overline{EF} \perp \overline{BC}$, $C$ is closer to $\overline{GH}$ than to $\overline{EF}$, and $D$ is closer to $\overline{FG}$ than to $\overline{EH}$, as shown. Triangles $\triangle {DGF}$ and $\triangle {CHG}$ have equal areas. The area of rectangle $EFGH$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

[asy] size(5cm); defaultpen(fontsize(10pt)); pair A = (9, 0), B = (15, 0), C = (-15, 0), D = (9, 12), E = (9+12/sqrt(5), -6/sqrt(5)), F = (9+12/sqrt(5), 6/sqrt(5)), G = (9-12/sqrt(5), 6/sqrt(5)), H = (9-12/sqrt(5), -6/sqrt(5)); filldraw(G--H--C--cycle, lightgray); filldraw(D--G--F--cycle, lightgray); draw(B--C); draw(A--D); draw(E--F--G--H--cycle); draw(circle(origin, 15)); draw(circle(A, 6)); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); label("$A$", A, (.8, -.8)); label("$B$", B, (.8, 0)); label("$C$", C, (-.8, 0)); label("$D$", D, (.4, .8)); label("$E$", E, (.8, -.8)); label("$F$", F, (.8, .8)); label("$G$", G, (-.8, .8)); label("$H$", H, (-.8, -.8)); label("$\omega_1$", (9, -5)); label("$\omega_2$", (-1, -13.5)); [/asy]

Solution 1 (Thorough)

Let $GH=2x$ and $GF=2y$. Notice that since $\overline{BC}$ is perpendicular to $\overline{GH}$ (can be proven using basic angle chasing) and $\overline{BC}$ is an extension of a diameter of $\omega_1$, then $\overline{CB}$ is the perpendicular bisector of $\overline{GH}$. Similarly, since $\overline{AD}$ is perpendicular to $\overline{GF}$ (also provable using basic angle chasing) and $\overline{AD}$ is part of a diameter of $\omega_1$, then $\overline{AD}$ is the perpendicular bisector of $\overline{GF}$.

From the Pythagorean Theorem on $\triangle GFH$, we have $(2x)^2+(2y)^2=12^2$, so $x^2+y^2=36$. To find our second equation for our system, we utilize the triangles given.

Let $I=\overline{GH}\cap\overline{CB}$. Then we know that $GFBI$ is also a rectangle since all of its angles can be shown to be right using basic angle chasing, so $FG=IB$. We also know that $CI+IB=2\cdot 15=30$. $IA=y$ and $AB=6$, so $CI=30-y-6=24-y$. Notice that $CI$ is a height of $\triangle CHG$, so its area is $\frac{1}{2}(2x)(24-y)=x(24-y)$.

Next, extend $\overline{DA}$ past $A$ to intersect $\omega_2$ again at $D'$. Since $\overline{BC}$ is given to be a diameter of $\omega_2$ and $\overline{BC}\perp\overline{AD}$, then $\overline{BC}$ is the perpendicular bisector of $\overline{DD'}$; thus $DA=D'A$. By Power of a Point, we know that $CA\cdot AB=DA\cdot AD'$. $CA=30-6=24$ and $AB=6$, so $DA\cdot AD'=(DA)^2=24\cdot6=144$ and $DA=D'A=12$.

Denote $J=\overline{DA}\cap\overline{GF}$. We know that $DJ=DA-AJ=12-x$ (recall that $GI=IH=x$, and it can be shown that $GIAJ$ is a rectangle). $\overline{DJ}$ is the height of $\triangle DGF$, so its area is $\frac{1}{2}(2y)(12-x)=y(12-x)$.

We are given that $[DGF]=[CHG]$ ($[ABC]$ denotes the area of figure $ABC$). As a result, $x(24-y)=y(12-x)$. This can be simplified to $y=2x$. Substituting this into the Pythagorean equation yields $5x^2=36$ and $x=\frac{6}{\sqrt{5}}$. Then $y=\frac{12}{\sqrt{5}}$.

$[EFGH]=2x\cdot2y=2\cdot\frac{6}{\sqrt{5}}\cdot2\cdot\frac{12}{\sqrt{5}}=\frac{288}{5}$, so the answer is $288+5=\boxed{293}$.

~ eevee9406

~ Edited by aoum

Solution 2 (Faster)

Denote the intersection of $BC$ and $w_1$ as $P$, the intersection of $BC$ and $GH$ be $Q$, and the center of $w_2$ to be $O$. Additionally, let $EF = GH = a, FG = EH = b$. We have that $CP = 18$ and $PQ = 6-\frac{b}{2}$. Considering right triangle $OAD$, $AD = 12$. Letting $R$ be the intersection of $AD$ and $FG$, $DR = 12 - \frac{b}{2}$. Using the equivalent area ratios: \[\frac{a(24-\frac{b}{2})}{2} = \frac{(12-\frac{a}{2})b}{2}\]

This equation gives $b=2a$. Using the Pythagorean Theorem on triangle $GHE$ gives that $a^2+b^2 = 144$. Plugging the reuslt $b=2a$ into this equation gives that the area of the triangle is $\frac{288}{5} \to \boxed{293}$.

~ Vivdax

~ Edited by aoum

~ Additional edits by fermat_sLastAMC

Solution 3 (Not Recommended)

You can use your ruler to check that $GF=2GH.$ Then, we have $GF^2+GH^2=36,$ and we solve the system of equations to get $\frac{288}{5},$ so the answer is $\boxed{293}.$

Note: This method is not recommended as the diagrams are not necessarily drawn to scale. However, it can be used in emergency situations or to verify the answer.

~derekwang2048


Solution 4 (Almost no Algebra)

If we draw segments connecting $A$ to $F, G,$ and $H,$ we can easily verify that all of the right triangles created are congruent. Thus, triangles $AGF$ and $AGH$ have equal areas, which means, by the given conditions, that kites $AGCH$ and $AGDF$ have equal areas. Thus, by the area formula for kites,

\[\frac{1}{2}(AD)(GF)=\frac{1}{2}(AC)(BH),\] or \[(AD)(GF)=(24)(GH).\]

Also, if we extend $AD$ to the other side of the large circle, the chord length formula gives

\[(AD)(AD)=(AC)(AB),\] so \[(AD)(AD)=(24)(6).\]


Squaring the second equation above gives \[(AD)(AD)(GF)(GF)=(24)(24)(GH)(GH).\] Dividing by the fourth equation gives \[(GF)^2=4(GH)^2.\] Since we know, by the Pythagorean Theorem, that $(GF)^2+(GH)^2=144,$ we can substitute to determine that $5(GH)^2=144.$

The desired area is $(GF)(GH)=2(GH)^2=\frac{2}{5}(144),$ which is $\frac{288}{5},$ so the answer is $\boxed{293}.$

See also

2025 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
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All AIME Problems and Solutions

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