Question

1. $x = \underline{}$

Explanation:

Step1: Identify tangent-secant theorem

For a tangent segment $TP$ and a secant segment $TA$ to a circle, the theorem states: $TP^2 = TX \times TA$, where $TX$ is the external part of the secant, and $TA$ is the full length of the secant. Here, $TX=6$, $TA = x+6$, $TP=12$.

Step2: Substitute values into theorem

$$12^2 = 6 \times (x + 6)$$

Step3: Simplify and solve for $x$

First calculate $12^2=144$, then:
$$144 = 6x + 36$$
Subtract 36 from both sides:
$$144 - 36 = 6x$$
$$108 = 6x$$
Divide both sides by 6:
$$x = \frac{108}{6} = 18$$

Answer:

18