Difference between revisions of "2005 AIME II Problems/Problem 8"

m (Solution)
(Solution)
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== Solution ==
 
== Solution ==
<center><asy>
+
<center>
pointpen = black; pathpen = black + linewidth(0.7); size(200);
+
<asy>
pair C1 = (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-acos(3/7));  
+
pointpen = black;  
 +
pathpen = black + linewidth(0.7);  
 +
size(200);
 +
 
 +
pair C1 = (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-arccos(3/7));  
 
path cir1 = CR(C1,4), cir2 = CR(C2,10), cir3 = CR(C3,14), t = H--T+2*(T-H);
 
path cir1 = CR(C1,4), cir2 = CR(C2,10), cir3 = CR(C3,14), t = H--T+2*(T-H);
 
pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);
 
pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);
D(cir1); D(cir2); D(cir3); D((14,0)--(-14,0)); D(A--B); D((-14,0)--D(MP("H",H,W))--A,linewidth(0.7) + linetype("4 4"));
+
 
D(MP("O_1",C1)); D(MP("O_2",C2)); D(MP("O_3",C3)); D(MP("T",T,N)); D(MP("A",A,NW)); D(MP("B",B,NE)); D(C1--MP("T_1",T1,N)); D(C2--MP("T_2",T2,N)); D(C3--T); D(rightanglemark(C_3,T,H));
+
draw(cir1); draw(cir2); draw(cir3);  
 +
draw((14,0)--(-14,0));
 +
draw(A--B);  
 +
draw((-14,0)--draw(MP("H",H,W))--A, linewidth(0.7) + linetype("4 4"));
 +
 
 +
draw(MP("O_1",C1));  
 +
draw(MP("O_2",C2));  
 +
draw(MP("O_3",C3));  
 +
 
 +
draw(MP("T",T,N));  
 +
draw(MP("A",A,NW));  
 +
draw(MP("B",B,NE));  
 +
draw(C1--MP("T_1",T1,N));  
 +
 
 +
draw(C2--MP("T_2",T2,N));  
 +
draw(C3--T);  
 +
draw(rightanglemark(C_3,T,H));
 
</asy></center>
 
</asy></center>
  

Revision as of 01:55, 20 December 2010

Problem

Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of the chord is $\frac{m\sqrt{n}}p$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p.$

Solution

pointpen = black; 
pathpen = black + linewidth(0.7); 
size(200);

pair C1 = (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-arccos(3/7)); 
path cir1 = CR(C1,4), cir2 = CR(C2,10), cir3 = CR(C3,14), t = H--T+2*(T-H);
pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);

draw(cir1); draw(cir2); draw(cir3); 
draw((14,0)--(-14,0));
draw(A--B); 
draw((-14,0)--draw(MP("H",H,W))--A, linewidth(0.7) + linetype("4 4"));

draw(MP("O_1",C1)); 
draw(MP("O_2",C2)); 
draw(MP("O_3",C3)); 

draw(MP("T",T,N)); 
draw(MP("A",A,NW)); 
draw(MP("B",B,NE)); 
draw(C1--MP("T_1",T1,N)); 

draw(C2--MP("T_2",T2,N)); 
draw(C3--T); 
draw(rightanglemark(C_3,T,H));
 (Error compiling LaTeX. 649373252d4173d611385686c3f42b8c4f0a770b.asy: 7.79: no matching variable 'arccos')

Let $O_1, O_2, O_3$ be the centers and $r_1 = 4, r_2 = 10,r_3 = 14$ the radii of the circles $C_1, C_2, C_3$. Let $T_1, T_2$ be the points of tangency from the common external tangent of $C_1, C_2$, respectively, and let the extension of $\overline{T_1T_2}$ intersect the extension of $\overline{O_1O_2}$ at a point $H$. Let the endpoints of the chord/tangent be $A,B$, and the foot of the perpendicular from $O_3$ to $\overline{AB}$ be $T$. From the similar right triangles $\triangle HO_1T_1 \sim \triangle HO_2T_2 \sim \triangle HO_3T$,

\[\frac{HO_1}{4} = \frac{HO_1+14}{10} = \frac{HO_1+10}{O_3T}.\]

It follows that $HO_1 = \frac{28}{3}$, and that $O_3T = \frac{58}{7}$. By the Pythagorean Theorem on $\triangle ATO_3$, we find that

\[AB = 2AT = 2\left(\sqrt{r_3^2 - O_3T^2}\right) = 2\sqrt{14^2 - \frac{58^2}{7^2}} = \frac{8\sqrt{390}}{7}\]

and the answer is $m+n+p=\boxed{405}$.

See also

2005 AIME II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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