Xavier Aptitude Test 2024 Paper

Consider a right-angled triangle $\mathit{A}\mathit{B}\mathit{C}$, right angled at $\mathit{B}$. Two circles, each of radius $\mathit{r}$, are drawn inside the triangle in such a way that one of them touches $\mathit{A}\mathit{B}$ and $\mathit{B}\mathit{C}$, while the other one touches $\mathit{A}\mathit{C}$ and $\mathit{B}\mathit{C}$. The two circles also touch each other (see the image below). If $\mathit{A}\mathit{B}=18\mathrm{cm}$ and $\mathit{B}\mathit{C}=24\mathrm{cm}$, then find the value of $\mathit{r}$.