Given Information:

- AB is the diameter of a circle, and its length is 5 cm.
- O is the center of the circle.
- The length of a tangent segment CT is 12 cm.

To determine CT, we can use the properties of tangents to circles.

Properties of Tangents to a Circle:

1. A line tangent to a circle is perpendicular to the radius drawn to the point of tangency.
2. The line segment joining the center of a circle to the point of tangency is perpendicular to the tangent line.

Solution:

Let's analyze the given figure to find the length of CT.

1. Draw the circle with center O and diameter AB.

2. Since AB is a diameter, we know that it passes through the center O. Thus, the line segment OB is a radius of the circle.

3. According to property 1, the tangent segment CT is perpendicular to the radius OB.

4. Now, we have a right-angled triangle OBC, where OC is the hypotenuse, OB is the perpendicular, and CT is the base.

5. We are given that AB = 5 cm, which is the diameter of the circle. Therefore, the radius OB is half the length of the diameter, i.e., OB = AB/2 = 5/2 = 2.5 cm.

6. Let's assume that CT = x cm.

7. By property 2, OC is perpendicular to CT. Therefore, OC is the height of the triangle OBC.

8. Using the Pythagorean theorem, we can find the length of OC:

OC^2 = OB^2 + BC^2
OC^2 = (2.5)^2 + x^2
OC^2 = 6.25 + x^2

9. Since OC is the hypotenuse and CT is the base, we can apply the Pythagorean theorem to find OC in terms of x:

OC^2 = CT^2 + OT^2
OC^2 = x^2 + OT^2

10. Since OC is the same in both equations (step 8 and step 9), we can equate them:

6.25 + x^2 = x^2 + OT^2

11. Simplifying the equation, we get:

6.25 = OT^2

12. Taking the square root on both sides, we find:

OT = √6.25 = 2.5 cm

13. Therefore, the length of CT is equal to the radius OT, which is 2.5 cm.

Thus, the length of the tangent segment CT is 2.5 cm.