3. In the diagram, $B D$ is perpendicular to $B C$ and to $A D$. If $A B=52, B C=21$, and $A D=48$, what is the length of $D C$ ?
3. Since $\triangle A D B$ and $\triangle C B D$ are right-angled, we can apply the Pythagorean Theorem in each triangle.

Therefore, $A B^{2}=A D^{2}+B D^{2}$ or $52^{2}=48^{2}+B D^{2}$ and so $B D^{2}=52^{2}-48^{2}=2704-2304=400$. Since $B D>0$, then $B D=\sqrt{400}=20$.
Furthermore, $D C^{2}=B D^{2}+B C^{2}=20^{2}+21^{2}=400+441=841$.
Since $D C>0$, then $D C=\sqrt{841}=29$.
ANSWER: $D C=29$