In the given figure, TA is a tangent to the circle with a center O such that OT is to four centimeter angle OTA is 30 degree, then the length of TA is.

In the given figure, we have a circle with center O. TA is a tangent to the circle at point A. We are given that OT is 4 centimeters and angle OTA is 30 degrees. We need to find the length of TA.

To solve this problem, we can use the properties of tangents to a circle.

Step 1: Draw a diagram representing the given information. Label the center of the circle as O, the point of tangency as A, and the point where the line OT intersects the circle as T.

Step 2: Since TA is a tangent to the circle, we know that the line TA is perpendicular to the radius OA at point A.

Step 3: Draw a right triangle OAT, where OA is the radius of the circle, OT is the given length of 4 centimeters, and angle OTA is 30 degrees.

Step 4: Since angle OTA is 30 degrees, we know that angle OAT is 90 - 30 = 60 degrees.

Step 5: Now, we can use trigonometry to find the length of TA. In right triangle OAT, we have the side opposite angle OAT (TA) and the side adjacent to angle OAT (OT).

Step 6: We can use the trigonometric function tangent (tan) to find the length of TA. The tangent of angle OAT is equal to the length of TA divided by the length of OT.

Step 7: Using the tangent function, we have tan(60 degrees) = TA / 4.

Step 8: Solving for TA, we have TA = 4 * tan(60 degrees).

Step 9: Using a calculator, we can find that tan(60 degrees) is approximately 1.732.

Step 10: Therefore, TA = 4 * 1.732 = 6.928 centimeters.

So, the length of TA is approximately 6.928 centimeters.