5. If the three roots of the equation $(x-1)\left(x^{2}-2 x+m\right)=0$ can serve as the lengths of the three sides of a triangle, then the range of the real number $m$ is ( ).
(A) $0<m \leqslant 1$
(B) $m \geqslant \frac{3}{4}$
(C) $\frac{3}{4}<m \leqslant 1$
(D) $m \geqslant 1$
5. C. Let $\alpha, \beta$ be the roots of the equation $x^{2}-2 x+m=0$. From the discriminant, we get $m \leqslant 1$. It is evident that $\alpha+\beta=2>1$. Also, by the triangle inequality, we have $|\alpha-\beta|<\frac{3}{4}$.