A triangle problem to calculate

By law of cosines

$44.76^2=71.22^2+73.94^2-2(71.22)(73.94) \cos C$

$\cos C=\dfrac{8989}{11091}$ $\implies$ $C= \cos^{-1}\left(\dfrac{8989}{11091}\right) \approx 35.86^\circ$

Consider right $\triangle AHC$ :

$\cos C=\dfrac{CH}{71.22}$ , however, $\cos C=\left(\dfrac{8989}{11091}\right)$ , so

$\dfrac{CH}{71.22}=\dfrac{8989}{11091}$ $\implies$ $CH \approx57.72$

Consider right $\triangle HDC$ :

$\sin C=\dfrac{DH}{CH}$

$\sin 35.86^\circ=\dfrac{DH}{57.72}$

$DH \approx$ $\boxed{33.81}$