3. in the diagram below of right triangle acb, altitudb find the length of cd in simplest radical form4,

ACB, ACD and DCB are all right triangles. We can then represent the relationships of their legs through the Pythagorean theorem. 

For triangle ACB:
7^2=(AC)^2+(BC)^2     (1)

For triangle ACD:
(AC)^2=3^2+(CD)^2     (2)

For triangle DCB:
(BC)^2=(CD)^2+4^2     (3)

To make things simpler, we let
(AC)=x
(BC)=y
(CD)=z

Hence we have:
49=x^2+y^2     (1)
y^2=z^2+16     (2)
x^2=9+z^2       (3)

To solve for the unknown c, let us first substitute equation (2) in (1)
49=x^2+z^2+16
33=x^2+z^2     (4)

Then, we can substitute equation (3) in (4)
33=9+z^2+z^2
24=2z^2
z^2=12
z=2sqrt(3)

Thus, the length of z or (CD) is 2sqrt(3) in simplest radical form.