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Difference between revisions of "2014 AIME II Problems/Problem 8"

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This problem can be very easily solved using Descartes' Circle Theorem. It states that if we have 4 circles that are all tangent with each other, <math>(k_1 + k_2 + k_3 + k_4)^{2} = 2(k_1^{2} + k_2^{2} + k_3^{2} + k_4^{2})</math>, where <math>k_i</math> is the curvature of circle <math>i</math>, meaning <math>k_i = \dfrac{1}{r}</math>. When three of the circles are internally tangent to the fourth one, the fourth circle has a negative curvature. Suppose we reflect Circle <math>E</math> over <math>\overline{AB}</math>. Now, we have our four circles to apply that theorem. First, lets scale our image down such that Circle <math>C</math> has radius <math>1</math>, for ease of computation. Let the radius of Circle <math>D</math> be <math>r</math>, so Circle <math>E</math> has radius <math>\dfrac{r}{3}</math>. Then, we have that <math>(-1 + \dfrac{1}{r} + \dfrac{3}{r} + \dfrac{3}{r})^{2} = 2(1 + \dfrac{1}{r^{2}} + \dfrac{9}{r^{2}} + \dfrac{9}{r^{2}})</math>. This simplifies to <math>\dfrac{49}{r^{2}} - \dfrac{14}{r} + 1 = \dfrac{2r^{2} + 38}{r^{2}}</math>. Multiplying both sides by <math>r^{2}</math>, we get that <math>49 - 14r + r^{2} = 2r^{2} + 38</math>, or <math>r^2 + 14r - 11 = 0</math>. We get <math>r = -7 \pm 2\sqrt{15}</math>, but we want the positive solution, which is <math>r = 2\sqrt{15} - 7</math>. We have to rescale back up, so we get <math>r = 4\sqrt{15} - 14 = \sqrt{240} - 14</math>, so we get that our answer is <math>240 + 14 = \boxed{254}</math>.
 
This problem can be very easily solved using Descartes' Circle Theorem. It states that if we have 4 circles that are all tangent with each other, <math>(k_1 + k_2 + k_3 + k_4)^{2} = 2(k_1^{2} + k_2^{2} + k_3^{2} + k_4^{2})</math>, where <math>k_i</math> is the curvature of circle <math>i</math>, meaning <math>k_i = \dfrac{1}{r}</math>. When three of the circles are internally tangent to the fourth one, the fourth circle has a negative curvature. Suppose we reflect Circle <math>E</math> over <math>\overline{AB}</math>. Now, we have our four circles to apply that theorem. First, lets scale our image down such that Circle <math>C</math> has radius <math>1</math>, for ease of computation. Let the radius of Circle <math>D</math> be <math>r</math>, so Circle <math>E</math> has radius <math>\dfrac{r}{3}</math>. Then, we have that <math>(-1 + \dfrac{1}{r} + \dfrac{3}{r} + \dfrac{3}{r})^{2} = 2(1 + \dfrac{1}{r^{2}} + \dfrac{9}{r^{2}} + \dfrac{9}{r^{2}})</math>. This simplifies to <math>\dfrac{49}{r^{2}} - \dfrac{14}{r} + 1 = \dfrac{2r^{2} + 38}{r^{2}}</math>. Multiplying both sides by <math>r^{2}</math>, we get that <math>49 - 14r + r^{2} = 2r^{2} + 38</math>, or <math>r^2 + 14r - 11 = 0</math>. We get <math>r = -7 \pm 2\sqrt{15}</math>, but we want the positive solution, which is <math>r = 2\sqrt{15} - 7</math>. We have to rescale back up, so we get <math>r = 4\sqrt{15} - 14 = \sqrt{240} - 14</math>, so we get that our answer is <math>240 + 14 = \boxed{254}</math>.
 
~Puck_0
 
~Puck_0
 +
 +
==Solution 5 (Heron's Formula)==
 +
This solution focuses on the area of <math>\triangle DEC</math>. Because <math>EF</math> is perpendicular to <math>AB</math>, it is an altitude of <math>\triangle DEC</math>. Therefore, we can express the area of <math>\triangle DEC</math> as <math>\frac{1}{2}\cdot EF \cdot DC</math>. We can also express the area of <math>\triangle DEC</math> using Heron's Formula.
 +
Let <math>r</math> equal the radius of circle <math>E</math>. Then <math>DC</math> = <math>CA - DA = 2-3r</math>. We also know that <math>CE = 2-r</math> and <math>DE = 3r+r=4r</math>.
 +
The semi-perimeter of <math>\triangle DEC</math> is <math>(DC+CE+DE)/2 = 2</math>.
 +
 +
Applying Heron's Formula, we get
 +
<cmath>\sqrt{2(2-4r)(r)(3r)} = \sqrt{6r^2(2-4r)}.</cmath>
 +
We set this equal to <math>\frac{1}{2}r(2-3r)</math>.
 +
<cmath>\sqrt{6r^2(2-4r)} = \frac{1}{2}r(2-3r).</cmath>
 +
This simplifies to the quadratic equation <math>0 = 9r^2+84r-44</math>. Remember that we are solving for <math>3r</math>, which we will set equal to <math>x</math>. Then we now have the equation <math>0 = x^2 +28r - 44</math>. Applying the quadratic formula, we get <math>x=-14 \pm 4\sqrt{15}</math>. We want the positive solution, so we take <math>4\sqrt{15}-14 = \sqrt{240}-14</math>. Our answer is therefore <math>240 + 14 = \boxed{254}</math>.
 +
 +
~lprado
  
 
== See also ==
 
== See also ==

Latest revision as of 19:56, 29 November 2024

Problem

Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $\sqrt{m}-n$, where $m$ and $n$ are positive integers. Find $m+n$.

Solution 1

[asy] import graph; size(7.99cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; real xmin = 4.087153740193288, xmax = 11.08175859031552, ymin = -4.938019122704778, ymax = 1.194137062512079; draw(circle((7.780000000000009,-1.320000000000002), 2.000000000000000)); draw(circle((7.271934046987930,-1.319179731427737), 1.491933384829670)); draw(circle((9.198812158392690,-0.8249788848962227), 0.4973111282761416)); draw((5.780002606580324,-1.316771019595571)--(9.779997393419690,-1.323228980404432)); draw((9.198812158392690,-0.8249788848962227)--(9.198009254448635,-1.322289365031666)); draw((7.271934046987930,-1.319179731427737)--(9.198812158392690,-0.8249788848962227)); draw((9.198812158392690,-0.8249788848962227)--(7.780000000000009,-1.320000000000002)); dot((7.780000000000009,-1.320000000000002),dotstyle); label("$C$", (7.707377218471464,-1.576266740352400), NE * labelscalefactor); dot((7.271934046987930,-1.319179731427737),dotstyle); label("$D$", (7.303064016111554,-1.276266740352400), NE * labelscalefactor); dot((9.198812158392690,-0.8249788848962227),dotstyle); label("$E$", (9.225301294671791,-0.7792624249832147), NE * labelscalefactor); dot((9.198009254448635,-1.322289365031666),dotstyle); label("$F$", (9.225301294671791,-1.276266740352400), NE * labelscalefactor); dot((9.779997393419690,-1.323228980404432),dotstyle); label("$B$", (9.810012253929656,-1.276266740352400), NE * labelscalefactor); dot((5.780002606580324,-1.316771019595571),dotstyle); label("$A$", (5.812051070003994,-1.276266740352400), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Using the diagram above, let the radius of $D$ be $3r$, and the radius of $E$ be $r$. Then, $EF=r$, and $CE=2-r$, so the Pythagorean theorem in $\triangle CEF$ gives $CF=\sqrt{4-4r}$. Also, $CD=CA-AD=2-3r$, so \[DF=DC+CF=2-3r+\sqrt{4-4r}.\] Noting that $DE=4r$, we can now use the Pythagorean theorem in $\triangle DEF$ to get \[(2-3r+\sqrt{4-4r})^2+r^2=16r^2.\]

Solving this quadratic is somewhat tedious, but the constant terms cancel, so the computation isn't terrible. Solving gives $3r=\sqrt{240}-14$ for a final answer of $\boxed{254}$.

  • Notice that C, E and the point of tangency to circle C for circle E will be collinear because C and E intersect the tangent line at a right angle, implying they must be on the same line.

Solution 2

Consider a reflection of circle $E$ over diameter $\overline{AB}$. By symmetry, we now have three circles that are pairwise externally tangent and all internally tangent to a large circle. The small circles have radii $r$, $r$, and $3r$, and the big circle has radius $2$.

Descartes' Circle Theorem gives $\left(\frac{1}{r}+\frac{1}{r}+\frac{1}{3r}-\frac12\right)^2 = 2\left(\left(\frac{1}{r}\right)^2+\left(\frac{1}{r}\right)^2+\left(\frac{1}{3r}\right)^2+\left(-\frac12\right)^2\right)$

Note that the big circle has curvature $-\frac12$ because it is internally tangent. Solving gives $3r=\sqrt{240}-14$ for a final answer of $\boxed{254}$.

Solution 3

We use the notation of Solution 1 for triangle $\triangle DEC$ \[\sin \angle EDC = \frac {EF}{DE} = \frac {1}{4} \implies \cos \angle EDC = \frac {\sqrt{15}}{4}.\] We use Cosine Law for $\triangle DEC$ and get: \[(4r)^2 +(2 – 3r)^2 – 2 \cdot 4r \cdot (2 – 3r) \cdot \frac {\sqrt{15}}{4} = (2 – r)^2\]. \[(24 + 6 \sqrt{15} ) r^2 = (8 + 4 \sqrt {15})r \implies 3r = 4 \sqrt{15} – 14 \implies \boxed{\textbf{254}}.\] vladimir.shelomovskii@gmail.com, vvsss

Solution 4

This problem can be very easily solved using Descartes' Circle Theorem. It states that if we have 4 circles that are all tangent with each other, $(k_1 + k_2 + k_3 + k_4)^{2} = 2(k_1^{2} + k_2^{2} + k_3^{2} + k_4^{2})$, where $k_i$ is the curvature of circle $i$, meaning $k_i = \dfrac{1}{r}$. When three of the circles are internally tangent to the fourth one, the fourth circle has a negative curvature. Suppose we reflect Circle $E$ over $\overline{AB}$. Now, we have our four circles to apply that theorem. First, lets scale our image down such that Circle $C$ has radius $1$, for ease of computation. Let the radius of Circle $D$ be $r$, so Circle $E$ has radius $\dfrac{r}{3}$. Then, we have that $(-1 + \dfrac{1}{r} + \dfrac{3}{r} + \dfrac{3}{r})^{2} = 2(1 + \dfrac{1}{r^{2}} + \dfrac{9}{r^{2}} + \dfrac{9}{r^{2}})$. This simplifies to $\dfrac{49}{r^{2}} - \dfrac{14}{r} + 1 = \dfrac{2r^{2} + 38}{r^{2}}$. Multiplying both sides by $r^{2}$, we get that $49 - 14r + r^{2} = 2r^{2} + 38$, or $r^2 + 14r - 11 = 0$. We get $r = -7 \pm 2\sqrt{15}$, but we want the positive solution, which is $r = 2\sqrt{15} - 7$. We have to rescale back up, so we get $r = 4\sqrt{15} - 14 = \sqrt{240} - 14$, so we get that our answer is $240 + 14 = \boxed{254}$. ~Puck_0

Solution 5 (Heron's Formula)

This solution focuses on the area of $\triangle DEC$. Because $EF$ is perpendicular to $AB$, it is an altitude of $\triangle DEC$. Therefore, we can express the area of $\triangle DEC$ as $\frac{1}{2}\cdot EF \cdot DC$. We can also express the area of $\triangle DEC$ using Heron's Formula. Let $r$ equal the radius of circle $E$. Then $DC$ = $CA - DA = 2-3r$. We also know that $CE = 2-r$ and $DE = 3r+r=4r$. The semi-perimeter of $\triangle DEC$ is $(DC+CE+DE)/2 = 2$.

Applying Heron's Formula, we get \[\sqrt{2(2-4r)(r)(3r)} = \sqrt{6r^2(2-4r)}.\] We set this equal to $\frac{1}{2}r(2-3r)$. \[\sqrt{6r^2(2-4r)} = \frac{1}{2}r(2-3r).\] This simplifies to the quadratic equation $0 = 9r^2+84r-44$. Remember that we are solving for $3r$, which we will set equal to $x$. Then we now have the equation $0 = x^2 +28r - 44$. Applying the quadratic formula, we get $x=-14 \pm 4\sqrt{15}$. We want the positive solution, so we take $4\sqrt{15}-14 = \sqrt{240}-14$. Our answer is therefore $240 + 14 = \boxed{254}$.

~lprado

See also

2014 AIME II (Problems • Answer Key • Resources)
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Problem 7 		Followed by
Problem 9
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