A right-angled triangle has an area of 10 units. The radius of the inscribed circle is 1 unit. What is the radius of the circumscribed circle around the triangle?
Given that the radius of the inscribed circle in a right-angled triangle is 1 unit, the legs can be written as \(a = 1 + x\) and \(b = 1 + y\).

![](https://cdn.mathpix.com/cropped/2024_05_02_65aaa4c9b13acb348870g-1.jpg?height=279&width=548&top_left_y=256&top_left_x=775)

The hypotenuse is \(c = x + y\) due to the equality of the tangent segments from the circle.

Write the area of the triangle as the sum of the areas of the smaller triangles formed by the inscribed circle:

\[
10 = \frac{1 + x}{2} + \frac{1 + y}{2} + \frac{x + y}{2}
\]

From this, we get \(x + y = 9\), which is also the diameter of the circumscribed circle, since the triangle is right-angled.

Thus, the desired radius is 4.5 units.