Let ABC be a triangle with side lengths 3, 4, 5. Calculate the radius of the inscribed circle.
If you didn't already know, by drawing the $3,4,5$ triangle you will realize that it is a right triangle:

![](https://cdn.mathpix.com/cropped/2024_05_10_732985225e470d5c7305g-033.jpg?height=667&width=851&top_left_y=439&top_left_x=623)

To calculate the radius $r$ of the inscribed circle, we will calculate the area $\mathcal{A}$ of the triangle in two different ways. First, $\mathcal{A}=\frac{3 \times 4}{2}=6$. Then, if we calculate the areas of the three differently colored triangles separately, we get:

$$
\mathcal{A}=\frac{5 \times \mathrm{r}}{2}+\frac{4 \times \mathrm{r}}{2}+\frac{3 \times \mathrm{r}}{2}=6 \mathrm{r}
$$

From this, we deduce that $r=1$.