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

Revision as of 00:52, 8 November 2019

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 $AR+BR+CR+DR$ ?

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

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 same 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 $\sqrt{(\frac{8}{5}\sqrt{x^2 + 24^2})^2 - 24^2}$ , 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, 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 $\boxed{D) 192}$ . ~Solution by hyxue

Number-Intensive Solution

Start by drawing $PQ$ first, because trying to get all four circles down will take you a few years. Next, because all circles have $P$ and $Q$ on them, and since all points on a circle are equidistant from the center, all circle centers lie on the perpendicular bisector of $PQ$ , and point $R$ is on this bisector.

In order for all the circle radii to be different (because the circles can't be congruent), two circle centers are on the same side of $PQ$ , and two are straddling it. For the latter two circles- just call them $A$ and $B$ - clearly $AR+BR$ is 39.

Now, let's take the next case. Then $C$ and $D$ lie on the same side. Construct the triangles from your picture, and use the Pythagorean Theorem (maybe divide all lengths by 3 to lessen big numbers) and then you get that the distance from $R$ to the closest circle center is $57$ . Therefore, the answer is $39+2*57+39=192 \boxed{(D)}$ .

Pythagorean Theorem Solution

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 $R$ . Let $AR=y$ and let $BP=x$ so that $AP=\frac{5}{8}x$ . Then, we have in right triangles $PBR$ and $PAR$ , $(39+y)^2+24^2=x^2$ $y^2+24^2=\frac{25}{64}x^2$ Subtracting the second from the first gives $x^2=64(2y+39)$ and substituting this in the second equation gives $y^2-50y-399=0=(y-57)(y+7).$ Since $y>0, y=57$ and $AR=57$ while $BR=AR+39=96.$ Since none of the circles are congruent, $C$ and $D$ must be on the opposite side of $R$ so $CR+DR=CD=39.$ Hence, $AR+BR+CR+DR=57+96+39=192,$ which is $\boxed{D}$ .

@@ Line 22: / Line 22: @@
 ==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{x^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, 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{D) 192}</math>.
+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{x^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, 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{D) 192}</math>.
 ~Solution by hyxue

2015 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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