Difference between revisions of "2023 AMC 12A Problems/Problem 18"

Revision as of 22:53, 9 November 2023

Problem

Circle $C_1$ and $C_2$ each have radius $1$ , and the distance between their centers is $\frac{1}{2}$ . Circle $C_3$ is the largest circle internally tangent to both $C_1$ and $C_2$ . Circle $C_4$ is internally tangent to both $C_1$ and $C_2$ and externally tangent to $C_3$ . What is the radius of $C_4$ ?

$[asy] import olympiad; size(10cm); draw(circle((0,0),0.75)); draw(circle((-0.25,0),1)); draw(circle((0.25,0),1)); draw(circle((0,6/7),3/28)); pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118); dot(B^^C); draw(B--E, dashed); draw(C--F, dashed); draw(B--C); label("$C_4$", D); label("$C_1$", (-1.375, 0)); label("$C_2$", (1.375,0)); label("$\frac{1}{2}$", (0, -.125)); label("$C_3$", (-0.4, -0.4)); label("$1$", (-.85, 0.70)); label("$1$", (.85, -.7)); import olympiad; markscalefactor=0.005; [/asy]$

$\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}$

Solution 1

With some simple geometry skills, we can find that $C_3$ has a radius of $\frac{3}{4}$ .

Since $C_4$ is internally tangent to $C_1$ , center of $C_4$ , $C_1$ and their tangent point must be on the same line.

Now, if we connect centers of $C_4$ , $C_3$ and $C_1$ / $C_2$ , we get a right angled triangle.

In which we get an equation by pythagorean theorem:

$(r+\frac{3}{4})^2+(\frac{1}{4})^2=(1-r)^2$

Solving it gives us

$r = \boxed{\textbf{(D) } \frac{3}{28}}$

~lptoggled

2023 AMC 10A (Problems • Answer Key • Resources)
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2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
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All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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Difference between revisions of "2023 AMC 12A Problems/Problem 18"

Revision as of 22:53, 9 November 2023

Problem

Solution 1

See Also

@@ Line 2: / Line 2: @@
 Circle <math>C_1</math> and <math>C_2</math> each have radius <math>1</math>, and the distance between their centers is <math>\frac{1}{2}</math>. Circle <math>C_3</math> is the largest circle internally tangent to both <math>C_1</math> and <math>C_2</math>. Circle <math>C_4</math> is internally tangent to both <math>C_1</math> and <math>C_2</math> and externally tangent to <math>C_3</math>. What is the radius of <math>C_4</math>?
-[someone pls insert diagram]
+<asy>
+import olympiad;
+size(10cm);
+draw(circle((0,0),0.75));
+draw(circle((-0.25,0),1));
+draw(circle((0.25,0),1));
+draw(circle((0,6/7),3/28));
+pair A = (0,0), B = (-0.25,0), C = (0.25,0), D = (0,6/7), E = (-0.95710678118, 0.70710678118), F = (0.95710678118, -0.70710678118);
+dot(B^^C);
+draw(B--E, dashed);
+draw(C--F, dashed);
+draw(B--C);
+label("$C_4$", D);
+label("$C_1$", (-1.375, 0));
+label("$C_2$", (1.375,0));
+label("$\frac{1}{2}$", (0, -.125));
+label("$C_3$", (-0.4, -0.4));
+label("$1$", (-.85, 0.70));
+label("$1$", (.85, -.7));
+import olympiad;
+markscalefactor=0.005;
+</asy>
 <math>\textbf{(A) } \frac{1}{14} \qquad \textbf{(B) } \frac{1}{12} \qquad \textbf{(C) } \frac{1}{10} \qquad \textbf{(D) } \frac{3}{28} \qquad \textbf{(E) } \frac{1}{9}</math>
@@ Line 25: / Line 46: @@
 ==See Also==
+{{AMC10 box|year=2023|ab=A|num-b=21|num-a=23}}
 {{AMC12 box|year=2023|ab=A|num-b=17|num-a=19}}
 {{MAA Notice}}