The length of [tex]CD[/tex] is 2.
How to determine the length of a segment
Since [tex]CD[/tex] is tangent to the circle, then [tex]AD\perp CD[/tex]. Then, the triangle [tex]ABD[/tex] is right angled. Thus, the length of [tex]CD[/tex] is found by Pythagorean theorem:
[tex]AC = \sqrt{AD^{2}+CD^{2}}[/tex] (1)
If we know that [tex]AB = 1.5[/tex], [tex]BC = 1[/tex] and [tex]AD = 1.5[/tex], then the length of [tex]CD[/tex] is:
[tex]CD = \sqrt{AC^{2}-AD^{2}}[/tex]
[tex]CD = \sqrt{2.5^{2}-1.5^{2}}[/tex]
[tex]CD = 2[/tex]
The length of [tex]CD[/tex] is 2. [tex]\blacksquare[/tex]
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