smc

Let $d_1=10$ and $d_2=24$ be the lengths of the diagonals, $a$ the side, and $r$ the radius of the inscribed circle. Using Pythagorean theorem we can compute $a=\sqrt{(d_1/2{)}^2+(d_2/2{)}^2}=13$. We can now express the area of the rhombus in two different ways: as $d_1{d}_2/2$, and as $2ar$. Solving $d_1{d}_2/2=2ar$ for $r$ we get $r=\boxed{\frac{60}{13}}$. (The first formula computes the area as one half of the circumscribed rectangle whose sides are parallel to the diagonals. The second one comes from the fact that we can divide the rhombus into $4$ equal triangles, and in those the height on the side $a$ is equal to $r$. See pictures below.)