Difference between revisions of "2007 AMC 10B Problems/Problem 18"

Revision as of 17:38, 25 February 2018

Problem

A circle of radius $1$ is surrounded by $4$ circles of radius $r$ as shown. What is $r$ ?

$[asy] unitsize(5mm); defaultpen(linewidth(.8pt)+fontsize(7pt)); dotfactor=4; real r1=1, r2=1+sqrt(2); pair A=(0,0), B=(1+sqrt(2),1+sqrt(2)), C=(-1-sqrt(2),1+sqrt(2)), D=(-1-sqrt(2),-1-sqrt(2)), E=(1+sqrt(2),-1-sqrt(2)); pair A1=(1,0), B1=(2+2sqrt(2),1+sqrt(2)), C1=(0,1+sqrt(2)), D1=(0,-1-sqrt(2)), E1=(2+2sqrt(2),-1-sqrt(2)); path circleA=Circle(A,r1); path circleB=Circle(B,r2); path circleC=Circle(C,r2); path circleD=Circle(D,r2); path circleE=Circle(E,r2); draw(circleA); draw(circleB); draw(circleC); draw(circleD); draw(circleE); draw(A--A1); draw(B--B1); draw(C--C1); draw(D--D1); draw(E--E1); label("$1$",midpoint(A--A1),N); label("$r$",midpoint(B--B1),N); label("$r$",midpoint(C--C1),N); label("$r$",midpoint(D--D1),N); label("$r$",midpoint(E--E1),N); [/asy]$

$\textbf{(A) } \sqrt{2} \qquad\textbf{(B) } 1+\sqrt{2} \qquad\textbf{(C) } \sqrt{6} \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 2+\sqrt{2}$

Solutions

[asy] unitsize(0.6cm); pair A,B,C,D; A=(-2.4,2.4); B=(2.4,2.4); C=(2.4,-2.4); D=(-2.4,-2.4); draw(A--B--C--D--cycle,linewidth(0.7)); draw(B--D,linewidth(0.7)); label(" $A$ ",A,NW); label(" $B$ ",B,NE); label(" $C$ ",C,SE); label(" $D$ ",D,SW); label("2",(0,0),SE); for (int i=0; i<2; ++i){ label(" $r$ ",(-2.4,-1.2+2.4i),W); label(" $r$ ",(-1.2+2.4i,2.4),N); label(" $r$ ",(-1.5+3i,-1.5+3i),NW); } for(int i=0; i<2; ++i){ for(int j=0; j<2; ++j){ draw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7)); }; } draw(Circle((0,0),1),linewidth(0.7)); [/asy]

Solution 1

You can express the line connecting the centers of an outer circle and the inner circle in two different ways. You can add the radius of both circles to get $r+1.$ You can also add the radius of two outer circles and use a $45-45-90$ triangle to get $\frac{2r}{\sqrt{2}} = r\sqrt{2}.$ Since both representations are for the same thing, you can set them equal to each other. $\begin{align*} r+1&=r\sqrt{2}\\ 1&=r(\sqrt{2}-1)\end{align*}$ $r = \frac{1}{\sqrt{2}-1} = \frac{\sqrt{2}+1}{2-1} = \boxed{\mathrm{(B) \ } 1 + \sqrt{2}}$

Solution 2

You can solve this problem by setting up a simple equation with the Pythagorean Theorem. The hypotenuse would be a segment that includes the radius of two circles on opposite corners and the diameter of the middle circle. This results in a segment of length $2r+2$ . The two legs are each the length between the centers of two large, adjacent circles, thus $2r$ . Using the Pythagorean Theorem: $\begin{align*} (2r+2)^2 = 2(2r)^2\\ 4r^2+8r+4=8r^2\\ r^2+2r+1=2r^2\\ r^2-2r-1=0\\ r=\frac{2+\sqrt{4-(-4)}}{2}=\frac{2+\sqrt{2}}{2}=\boxed{\mathrm{(B) \ } 1 + \sqrt{2}} \end{align*}$

2007 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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