Question

there is a circle in the figure, with a tangent line and a secant line from a point outside the circle. the angle between the tangent and the secant is 23 degrees. there are some numbers and expressions like 48, 48, and (14x - 22) around the circle.

Explanation:

Step1: Recall secant-tangent angle theorem

The measure of an angle formed by a tangent and a secant outside a circle is half the difference of the measures of the intercepted arcs.

Step2: Set up the equation

Let the intercepted arcs be $14x-22$ and $48$. The angle is $23^\circ$, so:
$$23 = \frac{1}{2}[(14x - 22) - 48]$$

Step3: Simplify the right-hand side

First combine terms inside the parentheses:
$$23 = \frac{1}{2}(14x - 70)$$

Step4: Multiply both sides by 2

Eliminate the fraction:
$$46 = 14x - 70$$

Step5: Solve for $x$

Add 70 to both sides, then divide by 14:
$$14x = 46 + 70 = 116$$
$$x = \frac{116}{14} = \frac{58}{7} \approx 8.29$$

Answer:

$x = \frac{58}{7}$ (or approximately $8.29$)