find the value of $x$.

Answer

Explanation:

Step1: Identify the relevant theorem

We use the secant - tangent rule. If a tangent and a secant are drawn to a circle from an exterior point, then the square of the length of the tangent is equal to the product of the lengths of the secant's external part and the entire secant. The length of the entire secant is (36 + 12=48), and the external part of the secant is 12.

Step2: Set up the equation

Let the length of the tangent be (x). According to the secant - tangent rule, (x^{2}=12\times(12 + 36)).

Step3: Calculate the right - hand side

First, calculate (12\times(12 + 36)=12\times48 = 576).

Step4: Solve for (x)

If (x^{2}=576), then (x=\sqrt{576}=24) (we take the positive value since length cannot be negative).