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.
