. A triangle has side lengths: 3, 4, 5. Calculate the radius of the inscribed circle (circle inside the triangle and tangent to all three sides of the triangle).
. Let $r$ be the radius of the inscribed circle. We start by drawing a figure.

![](https://cdn.mathpix.com/cropped/2024_05_10_55ca51a3a13dec3fb506g-13.jpg?height=446&width=560&top_left_y=1778&top_left_x=749)

The triangle $ABC$ with side lengths $3, 4, 5$ is a right triangle. We can calculate its area $S$ in two different ways: on the one hand, $S = \frac{4 \times 3}{2} = 6$; on the other hand, $S$ is the sum of the areas of triangles $IAB$, $IAC$, and $IBC$. The area of triangle $IAB$ is $\frac{r \times AB}{2} = 2r$, that of triangle $IAC$ is $\frac{r \times AC}{2} = \frac{3}{2}r$, and that of triangle $IBC$ is $\frac{r \times BC}{2} = \frac{5}{2}r$. We deduce that $6 = S = r \times \left(2 + \frac{3}{2} + \frac{5}{2}\right) = 6r$ and thus $r = 1$.