Question 25
Square $ABCD$ has side length $32$ cm. A square $A^\prime B^\prime C^\prime D^\prime$ is inscribed in $ABCD$ such that its vertices lie on the four sides of $ABCD$, with the point $A^\prime$ dividing side $AB$ in the ratio $AA^\prime:AB=1:4$. A second square $A^{\prime\prime}B^{\prime\prime}C^{\prime\prime}D^{\prime\prime}$ is then inscribed in $A^\prime B^\prime C^\prime D^\prime$ using the same construction, and a third square $A^{\prime\prime\prime}B^{\prime\prime\prime}C^{\prime\prime\prime}D^{\prime\prime\prime}$ is inscribed in $A^{\prime\prime}B^{\prime\prime}C^{\prime\prime}D^{\prime\prime}$ similarly. Find the total area of the shaded triangular regions lying between squares $ABCD$ and $A^\prime B^\prime C^\prime D^\prime$, and between squares $A^{\prime\prime}B^{\prime\prime}C^{\prime\prime}D^{\prime\prime}$ and $A^{\prime\prime\prime}B^{\prime\prime\prime}C^{\prime\prime\prime}D^{\prime\prime\prime}$ in $\mathrm{cm}^2$.
