Difference between revisions of "1997 AIME Problems/Problem 4"
(sol) |
|||
| Line 24: | Line 24: | ||
[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 18:35, 4 July 2013
Problem
Circles of radii 
and 
are mutually externally tangent, where 
and 
are relatively prime positive integers. Find 
Solution
If (in the diagram above) we draw the line going through the centers of the circles with radii 
and 
, that line is the perpendicular bisector of the segment connecting the centers of the two circles with radii 
. Then we form two right triangles, of lengths 
and 
, wher 
is the distance between the center of the circle in question and the segment connecting the centers of the two circles of radii . By the Pythagorean Theorem, we now have two equations with two unknowns:
$\begin{eqnarray*} 5^2 + x^2 &=& (5+r)^2 \\ x &=& \sqrt{10r + r^2} \\ && \\ (8 + r + \sqrt{10r+r^2})^2 + 5^2 &=& 13^2\\ 8 + r + \sqrt{10r+r^2} &=& 12\\ \sqrt{10r+r^2}&=& 4-r\\ 10r+r^2 &=& 16 - 8r + r^2\\ r &=& \frac{8}{9} \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)
So 
.
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
