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Difference between revisions of "2015 AMC 12B Problems/Problem 24"

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==Problem==
 
==Problem==
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Four circles, no two of which are congruent, have centers at <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, and points <math>P</math> and <math>Q</math> lie on all four circles. The radius of circle <math>A</math> is <math>\tfrac{5}{8}</math> times the radius of circle <math>B</math>, and the radius of circle <math>C</math> is <math>\tfrac{5}{8}</math> times the radius of circle <math>D</math>. Furthermore, <math>AB = CD = 39</math> and <math>PQ = 48</math>. Let <math>R</math> be the midpoint of <math>\overline{PQ}</math>. What is <math>AR+BR+CR+DR</math> ?
+
Four circles, no two of which are congruent, have centers at <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>, and points <math>P</math> and <math>Q</math> lie on all four circles. The radius of circle <math>A</math> is <math>\tfrac{5}{8}</math> times the radius of circle <math>B</math>, and the radius of circle <math>C</math> is <math>\tfrac{5}{8}</math> times the radius of circle <math>D</math>. Furthermore, <math>AB = CD = 39</math> and <math>PQ = 48</math>. Let <math>R</math> be the midpoint of <math>\overline{PQ}</math>. What is <math>\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}</math> ?
  
−
<math>\textbf{(A)}\; ? \qquad\textbf{(B)}\; ? \qquad\textbf{(C)}\; ? \qquad\textbf{(D)}\; ? \qquad\textbf{(E)}\; ?</math>
+
<math>\textbf{(A)}\; 180 \qquad\textbf{(B)}\; 184 \qquad\textbf{(C)}\; 188 \qquad\textbf{(D)}\; 192\qquad\textbf{(E)}\; 196</math>
  
 
==Solution==
 
==Solution==
 +
First, note that <math>PQ</math> lies on the radical axis of any of the pairs of circles. Suppose that <math>O_1</math> and <math>O_2</math> are the centers of two circles <math>C_1</math> and <math>C_2</math> that intersect exactly at <math>P</math> and <math>Q</math>, with <math>O_1</math> and <math>O_2</math> lying on the same side of <math>PQ</math>, and <math>O_1 O_2=39</math>. Let <math>x=O_1 R</math>, <math>y=O_2 R</math>, and suppose that the radius of circle <math>C_1</math> is <math>r</math> and the radius of circle <math>C_2</math> is <math>\tfrac{5}{8}r</math>.
  
 +
Then the power of point <math>R</math> with respect to <math>C_1</math> is
 +
 +
<cmath>(r+x)(r-x) = r^2 - x^2 = 24^2</cmath>
 +
 +
and the power of point <math>R</math> with respect to <math>C_2</math> is
 +
 +
<cmath>\left(\frac{5}{8}r + y\right) \left(\frac{5}{8}r - y\right) = \frac{25}{64}r^2 - y^2 = 24^2.</cmath>
 +
 +
Also, note that <math>x-y=39</math>.
 +
 +
Subtract the above two equations to find that <math>\tfrac{39}{64}r^2 - x^2 + y^2 = 0</math> or <math>39 r^2 = 64(x^2-y^2)</math>. As <math>x-y=39</math>, we find that <math>r^2=64(x+y) = 64(2y+39)</math>. Plug this into an earlier equation to find that <math>25(2y+39)-y^2=24^2</math>. This is a quadratic equation with solutions <math>y=\tfrac{50 \pm 64}{2}</math>, and as <math>y</math> is a length, it is positive, hence <math>y=57</math>, and <math>x=y+39=96</math>. This is the only possibility if the two centers lie on the same side of their radical axis.
 +
 +
On the other hand, if they lie on opposite sides, then it is clear that there is only one possibility, and then it is clear that <math>O_1 R + O_2 R = O_1 O_2 = 39</math>. Therefore, we obtain exactly four possible centers, and the sum of the desired lengths is <math>57+96+39 = \boxed{\textbf{(D)}\; 192}</math>.
 +
 +
==Solution 2==
 +
Note that if a circle passes through a pair of points, the center of the circle is on the perpendicular bisector of the line segment between the pair of points. This means that <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> are all on the perpendicular bisector of <math>PQ</math>. Let us say the distance from <math>A</math> to the line <math>PQ</math> is some <math>a</math>. Therefore, the distance from <math>B</math> to the line <math>PQ</math> is <math>\sqrt{(\frac{8}{5}\sqrt{a^2 + 24^2})^2 - 24^2}</math>, which comes out to be <math>\sqrt{\frac{64}{25}a^2 + \frac{39}{25}\cdot 576}</math>. Since <math>AB = 39</math>, we have one of <math>\sqrt{\frac{64}{25}a^2 + \frac{39}{25}\cdot 576} \pm a</math> to be equal to <math>39</math>. We can solve both equations to get that out of the four possible solutions, and only two are positive: <math>7</math> and <math>57</math>. Note that since no two circles can be congruent, we need the radius of one of <math>A</math> or <math>C</math> to be <math>7</math> and the other to be <math>57</math>. Plugging in to find the corresponding radii of <math>B</math> and <math>D</math> gives <math>32</math> and <math>96</math>, and adding everything up gives <math>\boxed{\textbf{(D) }192}</math>.
 +
 +
~hyxue
 +
 +
==Solution 3==
 +
Let's start by drawing <math>PQ</math>. Because all circles contain <math>P</math> and <math>Q</math>, all the centers lie on the perpendicular bisector of <math>PQ</math>, and point <math>R</math> is on this bisector.
 +
 +
For all the circle radii to be different (there can't be two congruent circles), two centers are on the same side of <math>PQ</math>, and two are on the opposite side of <math>PQ</math>. For the latter two circles--call them <math>A</math> and <math>B</math>--<math>AR+BR=39</math>.
 +
 +
Let's consider the next case, where <math>C</math> and <math>D</math> lie on the same side. Construct right triangles from the picture, and use the [[Pythagorean Theorem]] (divide by 3 to negate big numbers). You will get that the distance from <math>R</math> to the closest circle center is <math>57</math>. Therefore, the answer is <math>39+2\cdot57+39=\boxed{\textbf{(D) }192}</math>.
 +
 +
==Solution 4 (Pythagorean Theorem bash)==
 +
Since the radical axis <math>PQ</math> is perpendicular to the line connecting the center of the circles, we have that <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, and <math>R</math> are collinear. WLOG, assume that <math>A</math> and <math>B</math> are on the same side of <math>R</math> and let <math>AR=y</math> and let <math>BP=x</math> so that <math>AP=\frac{5}{8}x</math>.
 +
 +
Then, using the [[Pythagorean Theorem]] on right triangles <math>PBR</math> and <math>PAR</math>, 
 +
<cmath>(39+y)^2+24^2=x^2\qquad(1)</cmath>
 +
<cmath>y^2+24^2=\frac{25}{64}x^2\qquad(2)</cmath> Subtracting the <math>(2)</math> from <math>(1)</math> gives <cmath>x^2=64(2y+39)\qquad(3)</cmath>
 +
Substituting <math>(3)</math> into <math>(2)</math> gives <math>y^2-50y-399=0=(y-57)(y+7).</math>  Taking the positive solution (<math>y>0</math>), <math>y=AR=57</math> and <math>BR=(57)+39=96</math>.
 +
 +
Since none of the circles are congruent, <math>C</math> and <math>D</math> must be on the opposite side of <math>R</math> so <math>CR+DR=CD=39</math>. Hence, <math>AR+BR+CR+DR=57+96+39=\boxed{\textbf{(D) }192}</math>.
 +
 +
==Solution 5 (Fast)==
 +
Note that the four circles are coaxial, meaning <math>A,B,C,D,R</math> are all collinear. Let <math>AR=x</math>. By [[Pythagorean Theorem]], the radius of circle <math>A</math> squared would be
 +
<math>r_A^2 = x^2+24^2</math>
 +
and the radius of circle <math>B</math> squared would be
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<math>r_B^2 = (x+39)^2+24^2.</math>
 +
Since <math>r_A^2 = \dfrac{25}{64}r_B^2</math>,
 +
<cmath>x^2+24^2 = \dfrac{25}{64}((x+39)^2+24^2)</cmath>
 +
Solving this will give <math>x^2-50x-399 = 0</math>, or <math>x=57, -7</math>; <math>AR = 57</math>. The same equation will apply to <math>CR</math>; <math>CR</math> would be the other root: <math>CR=7</math> (the negative <math>7</math> signals that <math>C</math> is the negative, or opposite, direction of <math>D</math>, about <math>R</math>). Thus, <math>AR = 57</math>, <math>BR = 57+39 = 96</math>, <math>CR = 7</math>, <math>DR = 39-7 = 32</math>, implying
 +
<math>AR+BR+CR+DR = \boxed{\textbf{(D) }192}</math>.
 +
 +
~sml1809
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2015|ab=B|num-a=25|num-b=23}}
 
{{AMC12 box|year=2015|ab=B|num-a=25|num-b=23}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 18:18, 19 October 2023

Problem

Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\tfrac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. What is $\overline{AR}+\overline{BR}+\overline{CR}+\overline{DR}$ ?

$\textbf{(A)}\; 180 \qquad\textbf{(B)}\; 184 \qquad\textbf{(C)}\; 188 \qquad\textbf{(D)}\; 192\qquad\textbf{(E)}\; 196$

Solution

First, note that $PQ$ lies on the radical axis of any of the pairs of circles. Suppose that $O_1$ and $O_2$ are the centers of two circles $C_1$ and $C_2$ that intersect exactly at $P$ and $Q$, with $O_1$ and $O_2$ lying on the same side of $PQ$, and $O_1 O_2=39$. Let $x=O_1 R$, $y=O_2 R$, and suppose that the radius of circle $C_1$ is $r$ and the radius of circle $C_2$ is $\tfrac{5}{8}r$.

Then the power of point $R$ with respect to $C_1$ is

\[(r+x)(r-x) = r^2 - x^2 = 24^2\]

and the power of point $R$ with respect to $C_2$ is

\[\left(\frac{5}{8}r + y\right) \left(\frac{5}{8}r - y\right) = \frac{25}{64}r^2 - y^2 = 24^2.\]

Also, note that $x-y=39$.

Subtract the above two equations to find that $\tfrac{39}{64}r^2 - x^2 + y^2 = 0$ or $39 r^2 = 64(x^2-y^2)$. As $x-y=39$, we find that $r^2=64(x+y) = 64(2y+39)$. Plug this into an earlier equation to find that $25(2y+39)-y^2=24^2$. This is a quadratic equation with solutions $y=\tfrac{50 \pm 64}{2}$, and as $y$ is a length, it is positive, hence $y=57$, and $x=y+39=96$. This is the only possibility if the two centers lie on the same side of their radical axis.

On the other hand, if they lie on opposite sides, then it is clear that there is only one possibility, and then it is clear that $O_1 R + O_2 R = O_1 O_2 = 39$. Therefore, we obtain exactly four possible centers, and the sum of the desired lengths is $57+96+39 = \boxed{\textbf{(D)}\; 192}$.

Solution 2

Note that if a circle passes through a pair of points, the center of the circle is on the perpendicular bisector of the line segment between the pair of points. This means that $A$, $B$, $C$, and $D$ are all on the perpendicular bisector of $PQ$. Let us say the distance from $A$ to the line $PQ$ is some $a$. Therefore, the distance from $B$ to the line $PQ$ is $\sqrt{(\frac{8}{5}\sqrt{a^2 + 24^2})^2 - 24^2}$, which comes out to be $\sqrt{\frac{64}{25}a^2 + \frac{39}{25}\cdot 576}$. Since $AB = 39$, we have one of $\sqrt{\frac{64}{25}a^2 + \frac{39}{25}\cdot 576} \pm a$ to be equal to $39$. We can solve both equations to get that out of the four possible solutions, and only two are positive: $7$ and $57$. Note that since no two circles can be congruent, we need the radius of one of $A$ or $C$ to be $7$ and the other to be $57$. Plugging in to find the corresponding radii of $B$ and $D$ gives $32$ and $96$, and adding everything up gives $\boxed{\textbf{(D) }192}$.

~hyxue

Solution 3

Let's start by drawing $PQ$. Because all circles contain $P$ and $Q$, all the centers lie on the perpendicular bisector of $PQ$, and point $R$ is on this bisector.

For all the circle radii to be different (there can't be two congruent circles), two centers are on the same side of $PQ$, and two are on the opposite side of $PQ$. For the latter two circles--call them $A$ and $B$--$AR+BR=39$.

Let's consider the next case, where $C$ and $D$ lie on the same side. Construct right triangles from the picture, and use the Pythagorean Theorem (divide by 3 to negate big numbers). You will get that the distance from $R$ to the closest circle center is $57$. Therefore, the answer is $39+2\cdot57+39=\boxed{\textbf{(D) }192}$.

Solution 4 (Pythagorean Theorem bash)

Since the radical axis $PQ$ is perpendicular to the line connecting the center of the circles, we have that $A$, $B$, $C$, $D$, and $R$ are collinear. WLOG, assume that $A$ and $B$ are on the same side of $R$ and let $AR=y$ and let $BP=x$ so that $AP=\frac{5}{8}x$.

Then, using the Pythagorean Theorem on right triangles $PBR$ and $PAR$, \[(39+y)^2+24^2=x^2\qquad(1)\] \[y^2+24^2=\frac{25}{64}x^2\qquad(2)\] Subtracting the $(2)$ from $(1)$ gives \[x^2=64(2y+39)\qquad(3)\] Substituting $(3)$ into $(2)$ gives $y^2-50y-399=0=(y-57)(y+7).$ Taking the positive solution ($y>0$), $y=AR=57$ and $BR=(57)+39=96$.

Since none of the circles are congruent, $C$ and $D$ must be on the opposite side of $R$ so $CR+DR=CD=39$. Hence, $AR+BR+CR+DR=57+96+39=\boxed{\textbf{(D) }192}$.

Solution 5 (Fast)

Note that the four circles are coaxial, meaning $A,B,C,D,R$ are all collinear. Let $AR=x$. By Pythagorean Theorem, the radius of circle $A$ squared would be $r_A^2 = x^2+24^2$ and the radius of circle $B$ squared would be $r_B^2 = (x+39)^2+24^2.$ Since $r_A^2 = \dfrac{25}{64}r_B^2$, \[x^2+24^2 = \dfrac{25}{64}((x+39)^2+24^2)\] Solving this will give $x^2-50x-399 = 0$, or $x=57, -7$; $AR = 57$. The same equation will apply to $CR$; $CR$ would be the other root: $CR=7$ (the negative $7$ signals that $C$ is the negative, or opposite, direction of $D$, about $R$). Thus, $AR = 57$, $BR = 57+39 = 96$, $CR = 7$, $DR = 39-7 = 32$, implying $AR+BR+CR+DR = \boxed{\textbf{(D) }192}$.

~sml1809

See Also

2015 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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