35. In the following diagram, $A B \perp B C . D$ and $E$ are points on segments $A B$ and $B C$ respectively, such that $B A+A E=B D+D C$. It is known that $A D=2, B E=3$ and $E C=4$. Find $B A+A E$.
35. Answer: 10.
By given, $B D+2+A E=B D+D C$. So $2+A E=D C$.
Note that $A B^{2}+B E^{2}=A E^{2}$ and $B D^{2}+B C^{2}=D C^{2}$. Then
$$
\begin{array}{l}
(2+B D)^{2}+3^{2}=A E^{2}, \quad B D^{2}+7^{2}=(A E+2)^{2} \\
4(A E+B D)=32 \text {. Then } A E+B A=\frac{32}{4}+2=10 \text {. } \\
\end{array}
$$