Difference between revisions of "2005 AIME II Problems/Problem 8"
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pair C1 = (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-arccos(3/7)); | 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(C1,4.01), 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); | ||
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Let <math>O_1, O_2, O_3</math> be the centers and <math>r_1 = 4, r_2 = 10,r_3 = 14</math> the radii of the circles <math>C_1, C_2, C_3</math>. Let <math>T_1, T_2</math> be the points of tangency from the common external tangent of <math>C_1, C_2</math>, respectively, and let the extension of <math>\overline{T_1T_2}</math> intersect the extension of <math>\overline{O_1O_2}</math> at a point <math>H</math>. Let the endpoints of the chord/tangent be <math>A,B</math>, and the foot of the perpendicular from <math>O_3</math> to <math>\overline{AB}</math> be <math>T</math>. From the similar [[right triangle]]s <math>\triangle HO_1T_1 \sim \triangle HO_2T_2 \sim \triangle HO_3T </math>, | Let <math>O_1, O_2, O_3</math> be the centers and <math>r_1 = 4, r_2 = 10,r_3 = 14</math> the radii of the circles <math>C_1, C_2, C_3</math>. Let <math>T_1, T_2</math> be the points of tangency from the common external tangent of <math>C_1, C_2</math>, respectively, and let the extension of <math>\overline{T_1T_2}</math> intersect the extension of <math>\overline{O_1O_2}</math> at a point <math>H</math>. Let the endpoints of the chord/tangent be <math>A,B</math>, and the foot of the perpendicular from <math>O_3</math> to <math>\overline{AB}</math> be <math>T</math>. From the similar [[right triangle]]s <math>\triangle HO_1T_1 \sim \triangle HO_2T_2 \sim \triangle HO_3T </math>, | ||
− | <cmath>\frac{HO_1}{4} = \frac{HO_1+14}{10} = \frac{HO_1+10}{O_3T}.</cmath> | + | <cmath> \frac{HO_1}{4} = \frac{HO_1+14}{10} = \frac{HO_1+10}{O_3T}. </cmath> |
It follows that <math>HO_1 = \frac{28}{3}</math>, and that <math>O_3T = \frac{58}{7}</math>. By the [[Pythagorean Theorem]] on <math>\triangle ATO_3</math>, we find that | It follows that <math>HO_1 = \frac{28}{3}</math>, and that <math>O_3T = \frac{58}{7}</math>. By the [[Pythagorean Theorem]] on <math>\triangle ATO_3</math>, we find that |
Revision as of 02:49, 20 December 2010
Problem
Circles and are externally tangent, and they are both internally tangent to circle The radii of and are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of is also a common external tangent of and Given that the length of the chord is where and are positive integers, and are relatively prime, and is not divisible by the square of any prime, find
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(C1,4.01), 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. 8e782f834de02d6d9a3cce2457488eb56ee2d662.asy: 7.79: no matching variable 'arccos')
Let be the centers and the radii of the circles . Let be the points of tangency from the common external tangent of , respectively, and let the extension of intersect the extension of at a point . Let the endpoints of the chord/tangent be , and the foot of the perpendicular from to be . From the similar right triangles ,
It follows that , and that . By the Pythagorean Theorem on , we find that
and the answer is .
See also
2005 AIME II (Problems • Answer Key • Resources) | ||
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 |