There's probably a ton of ways to do this. Are you looking for a clever answer to this by any chance?

A straightforward approach would be to use

the law of cosines, which will give you all the angles in the triangles from their lengths. If you assume a is at the origin and that b is on the x axis an appropiate distance away from a, then you can use basic trig to find vector coordinates of all the points in that embedding. Once you do this, you can find

the equation of the plane and then

solve for the distance from point d to that plane.

Alternatively, you could first find the volume of the tetrahedron ABCD using the Cayley-Menger determinant (I didn't know about this until I just read it). You can also find the are of the triangle ABC using Heron's formula. The area of a tetrahedron relates these to each other using the distance between d and the triangle ABC:

(Volume of the tetrahedron ABCD) = 1/3 * (distance from d to triangle ABC) * (Area of Triangle ABC)