Example 1. In $\triangle A B C$, $B C=3, A C=4, A E$ and $B D$ are the medians to sides $B C$ and $A C$ respectively, and $A E \perp B D$. Then the length of $A B$ is ( ).
(A) $\sqrt{5}$
(B) $3 \sqrt{6}$
(C) $2 \sqrt{2}$
(D) $\sqrt{7}$
(E) cannot be determined
Given that $AE$ and $BD$ are medians of $\triangle ABC$, and $AE \perp BD$, according to property $3^{\circ}$, we know
$$
AC^{2} + BC^{2} = 5 AB^{2} \text{.}
$$

Therefore, $AB = \sqrt{\frac{AC^{2} + BC^{2}}{5}} = \sqrt{5}$. Hence, the correct choice is (A).