3. As shown in Figure 1, on the hypotenuse $AB$ of isosceles right $\triangle ABC$, take two points $M$ and $N$ (not coinciding with points $A$ and $B$) such that $\angle MCN=45^{\circ}$. Let $AM=a, MN=b, NB=c$. Then the shape of the triangle with side lengths $a, b, c$ is ( ).
(A) Acute triangle
(B) Obtuse triangle
(C) Right triangle
(D) Cannot be determined
3. C.
As shown in Figure 8, rotate $\triangle A C M$ counterclockwise around point $C$ by $90^{\circ}$, so that $A C$ coincides with $B C$, resulting in $\triangle B C M^{\prime}$. Connect $N M^{\prime}$. Then
$$
\begin{array}{l}
\triangle N C M^{\prime} \cong \triangle N C M \\
\Rightarrow N M^{\prime}=N M=b . \\
\text { Also, } \angle N B M^{\prime}=\angle N B C+\angle C A M=90^{\circ},
\end{array}
$$

Thus, the shape of the triangle with side lengths $a, b, c$ is a right triangle.