Difference between revisions of "2001 AMC 12 Problems/Problem 18"
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=== Solution 3 (Basically 1 but less complicated) === | === Solution 3 (Basically 1 but less complicated) === | ||
As in solution 1, in triangle <math>ABC</math> we have <math>AB = 1+4 = 5</math> and <math>BC=4-1 = 3</math>, thus by the Pythagorean theorem or pythagorean triples in general, we have <math>AC=4</math>. | As in solution 1, in triangle <math>ABC</math> we have <math>AB = 1+4 = 5</math> and <math>BC=4-1 = 3</math>, thus by the Pythagorean theorem or pythagorean triples in general, we have <math>AC=4</math>. | ||
− | Let <math>r</math> be the radius. Let <math>s</math> be the perpendicular intersecting point <math>S</math> and line <math>BC</math>. <math>AC=s</math> because <math>s,</math> both perpendicular radii, and <math>AC</math> form a rectangle. We just have to find <math>AC</math> in terms of <math>r</math> and solve for <math>r</math> now. | + | Let <math>r</math> be the radius. Let <math>s</math> be the perpendicular intersecting point <math>S</math> and line <math>BC</math>. <math>AC=s</math> because <math>s,</math> both perpendicular radii, and <math>AC</math> form a rectangle. We just have to find <math>AC</math> in terms of <math>r</math> and solve for <math>r</math> now. From the Pythagorean theorem and subtracting to get lengths, we get <math>AC=4=\sqrt{(r+1)^2 - (1-r)^2} + \sqrt{(r+4)^2 - (4-r)^2}</math>, which is simply $4=\sqrt{4r}+\sqrt{16r} \implies \sqrt{r}=\frac{2}{3} \implies r= \boxed{ |
=== Video Solution === | === Video Solution === |
Revision as of 01:24, 9 January 2024
Contents
Problem
A circle centered at with a radius of 1 and a circle centered at with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle?
Solution
Solution 1
In the triangle we have and , thus by the Pythagorean theorem we have .
Let be the radius of the small circle, and let be the perpendicular distance from to . Moreover, the small circle is tangent to both other circles, hence we have and .
We have and . Hence we get the following two equations:
Simplifying both, we get
As in our case both and are positive, we can divide the second one by the first one to get .
Now there are two possibilities: either , or .
In the first case clearly , which puts the center on the wrong side of , so this is not the correct case.
(Note: This case corresponds to the other circle that is tangent to both given circles and the common tangent line. By coincidence, due to the ratio between radii of and , this circle turns out to have the same radius as circle , with center directly left of center , and tangent to directly above center .)
The second case solves to . We then have , hence .
More generally, for two large circles of radius and , the radius of the small circle is .
Equivalently, we have that .
Solution 2
The horizontal line is the equivalent of a circle of curvature , thus we can apply Descartes' Circle Formula.
The four circles have curvatures , and .
We have
Simplifying, we get
Obviously cannot equal , therefore .
Solution 3 (Basically 1 but less complicated)
As in solution 1, in triangle we have and , thus by the Pythagorean theorem or pythagorean triples in general, we have . Let be the radius. Let be the perpendicular intersecting point and line . because both perpendicular radii, and form a rectangle. We just have to find in terms of and solve for now. From the Pythagorean theorem and subtracting to get lengths, we get , which is simply $4=\sqrt{4r}+\sqrt{16r} \implies \sqrt{r}=\frac{2}{3} \implies r= \boxed{
Video Solution
See Also
2001 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.