The figure shows three circles with radii 1, 2, and 3, tangent to each other at the highlighted points. What is the length of the segment $A B$?

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-13.jpg?height=351&width=508&top_left_y=1975&top_left_x=680)
A) 1
B) $\sqrt{2}$
C) $\frac{1+\sqrt{5}}{2}$
D) $\frac{3}{2}$
E) $\sqrt{3}$

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Solution

## ALTERNATIVE B

First, we recall that if two circles are tangent, then the line passing through their centers also passes through the point of tangency. In our case, calling the centers of the circles $P, Q$, and $R$ (as in the figure), this shows that $P R=3, P Q=4$, and $Q R=5$. Since $3^{2}+4^{2}=5^{2}$, it follows that triangle $P Q R$ is a right triangle at $P$. Additionally, since $P A=P B=1$, we see that $A B$ is the diagonal of a square with side length 1, i.e., $A B=\sqrt{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_01_c051a4aaadd76ab6970bg-43.jpg?height=376&width=554&top_left_y=246&top_left_x=660)

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