3 [Pythagorean Theorem (direct and inverse)]

Three circles of different radii touch each other externally. The segments connecting their centers form a right triangle. Find the radius of the smallest circle if the radii of the largest and medium circles are 6 and 4.

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Let $x$ be the radius of the smaller circle. The sides of the resulting triangle are $10, 6+x$, and $4+x$. Since 10 is the largest side, it is the hypotenuse. Therefore, $(x+6)^{2}+(x+4)^{2}=100$, from which $x=2$.

![](https://cdn.mathpix.com/cropped/2024_05_06_0ef0b80dacb6d98eff5ag-08.jpg?height=392&width=583&top_left_y=2005&top_left_x=746)

## Answer