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Difference between revisions of "2001 AMC 12 Problems/Problem 18"

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m (→‎Solution 1: removed coordinate clutter, added general formula.)
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In the triangle <math>ABC</math> we have <math>AB = 1+4 = 5</math> and <math>BC=4-1 = 3</math>, thus by the [[Pythagorean theorem]] we have <math>AC=4</math>.
 
In the triangle <math>ABC</math> we have <math>AB = 1+4 = 5</math> and <math>BC=4-1 = 3</math>, thus by the [[Pythagorean theorem]] we have <math>AC=4</math>.
  
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We can now pick a coordinate system where the common tangent is the <math>x</math> axis and <math>A</math> lies on the <math>y</math> axis.
 
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In this coordinate system we have <math>A=(0,1)</math> and <math>B=(4,4)</math>.
 
  
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Let <math>r</math> be the radius of the small circle, and let <math>s</math> be the <math>x</math>-coordinate of its center <math>S</math>. We then know that <math>S=(s,r)</math>, as the circle is tangent to the <math>x</math> axis. Moreover, the small circle is tangent to both other circles, hence we have <math>SA=1+r</math> and <math>SB=4+r</math>.
+
Let <math>r</math> be the radius of the small circle, and let <math>s</math> be the perpendicular distance from <math>A</math> to <math>\overline{BC}. Moreover, the small circle is tangent to both other circles, hence we have </math>SA=1+r<math> and </math>SB=4+r<math>.
  
−
We have <math>SA = \sqrt{s^2 + (1-r)^2}</math> and <math>SB=\sqrt{(4-s)^2 + (4-r)^2}</math>. Hence we get the following two equations:
+
We have </math>SA = \sqrt{s^2 + (1-r)^2}<math> and </math>SB=\sqrt{(4-s)^2 + (4-r)^2}<math>. Hence we get the following two equations:
  
 
<cmath>
 
<cmath>
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</cmath>
 
</cmath>
  
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As in our case both <math>r</math> and <math>s</math> are positive, we can divide the second one by the first one to get <math>\left( \frac{4-s}s \right)^2 = 4</math>.
+
As in our case both </math>r<math> and </math>s<math> are positive, we can divide the second one by the first one to get </math>\left( \frac{4-s}s \right)^2 = 4<math>.
  
−
Now there are two possibilities: either <math>\frac{4-s}s=-2</math>, or <math>\frac{4-s}s=2</math>. In the first case clearly <math>s<0</math>, hence this is not the correct case. (Note: This case corresponds to the other circle that is tangent to both given circles and the <math>x</math> axis - a large circle whose center is somewhere to the left of <math>A</math>.) The second case solves to <math>s=\frac 43</math>. We then have <math>4r = s^2 = \frac {16}9</math>, hence <math>r = \boxed{\frac 49}</math>.
+
Now there are two possibilities: either </math>\frac{4-s}s=-2<math>, or </math>\frac{4-s}s=2<math>. In the first case clearly </math>s<0<math>, hence 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 - a large circle whose center is somewhere to the left of </math>A<math>.) The second case solves to </math>s=\frac 43<math>. We then have </math>4r = s^2 = \frac {16}9<math>, hence </math>r = \boxed{\frac 49}<math>.
 +
 
 +
 
 +
More generally, for two large circles of radius </math>a<math> and </math>b<math>, the radius of the small circle is </math>\frac{ab}{(a+b)^2}$
  
 
=== Solution 2 ===
 
=== Solution 2 ===

Revision as of 22:56, 19 August 2023

Problem

A circle centered at $A$ with a radius of 1 and a circle centered at $B$ 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?

[asy] unitsize(0.75cm); pair A=(0,1), B=(4,4); dot(A); dot(B); draw( circle(A,1) ); draw( circle(B,4) ); draw( (-1.5,0)--(8.5,0) ); draw( A -- (A+(-1,0)) ); label("$1$", A -- (A+(-1,0)), N ); draw( B -- (B+(4,0)) ); label("$4$", B -- (B+(4,0)), N ); label("$A$",A,E); label("$B$",B,W); filldraw( circle( (12/9,4/9), 4/9 ), lightgray, black ); dot( (12/9,4/9) ); [/asy]

$\text{(A) }\frac {1}{3} \qquad \text{(B) }\frac {2}{5} \qquad \text{(C) }\frac {5}{12} \qquad \text{(D) }\frac {4}{9} \qquad \text{(E) }\frac {1}{2}$

Solution

Solution 1

[asy] unitsize(1cm); pair A=(0,1), B=(4,4), C=(4,1), S=(12/9,4/9); dot(A); dot(B); draw( circle(A,1) ); draw( circle(B,4) ); draw( (-1.5,0)--(8.5,0) ); draw( (A+(4,0)) -- A -- (A+(0,-1)) ); draw( A -- B -- (B+(0,-4)) ); label("$A$",A,N); label("$B$",B,N); label("$C$",C,E); label("$S$",S,N); filldraw( circle(S,4/9), lightgray, black ); dot(S); draw( rightanglemark(A,C,B) ); draw( S -- A ); draw( S -- B ); [/asy]

In the triangle $ABC$ we have $AB = 1+4 = 5$ and $BC=4-1 = 3$, thus by the Pythagorean theorem we have $AC=4$.


Let $r$ be the radius of the small circle, and let $s$ be the perpendicular distance from $A$ to $\overline{BC}. Moreover, the small circle is tangent to both other circles, hence we have$SA=1+r$and$SB=4+r$.

We have$ (Error compiling LaTeX. Unknown error_msg)SA = \sqrt{s^2 + (1-r)^2}$and$SB=\sqrt{(4-s)^2 + (4-r)^2}$. Hence we get the following two equations:

<cmath> \begin{align*} s^2 + (1-r)^2 & = (1+r)^2 \\ (4-s)^2 + (4-r)^2 & = (4+r)^2 \end{align*} </cmath>

Simplifying both, we get

<cmath> \begin{align*} s^2 & = 4r \\ (4-s)^2 & = 16r \end{align*} </cmath>

As in our case both$ (Error compiling LaTeX. Unknown error_msg)r$and$s$are positive, we can divide the second one by the first one to get$\left( \frac{4-s}s \right)^2 = 4$.

Now there are two possibilities: either$ (Error compiling LaTeX. Unknown error_msg)\frac{4-s}s=-2$, or$\frac{4-s}s=2$. In the first case clearly$s<0$, hence this is not the correct case. (Note: This case corresponds to the other circle that is tangent to both given circles and the common tangent line - a large circle whose center is somewhere to the left of$A$.) The second case solves to$s=\frac 43$. We then have$4r = s^2 = \frac {16}9$, hence$r = \boxed{\frac 49}$.


More generally, for two large circles of radius$ (Error compiling LaTeX. Unknown error_msg)a$and$b$, the radius of the small circle is$\frac{ab}{(a+b)^2}$

Solution 2

The horizontal line is the equivalent of a circle of curvature $0$, thus we can apply Descartes' Circle Formula.

The four circles have curvatures $0, 1, \frac 14$, and $\frac 1r$.

We have $2\left(0^2+1^2+\frac {1}{4^2}+\frac{1}{r^2}\right)=\left(0+1+\frac 14+\frac 1r\right)^2$

Simplifying, we get $\frac{34}{16}+\frac{2}{r^2}=\frac{25}{16}+\frac{5}{2r}+\frac{1}{r^2}$

\[\frac{1}{r^2}-\frac{5}{2r}+\frac{9}{16}=0\] \[\frac{16}{r^2}-\frac{40}{r}+9=0\] \[\left(\frac{4}{r}-9\right)\left(\frac{4}{r}-1\right)=0\]

Obviously $r$ cannot equal $4$, therefore $r = \boxed{\frac 49}$.

Video Solution

https://youtu.be/zOwYoFOUg2U

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

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