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In this problem, you are given an angle formed by a tangent and a secant intersecting outside a circle. The measures involved are:

- The external angle formed by the tangent and the secant, given as \( 23^\circ \),
- The arc subtended by the secant, labeled as \( (14x - 22)^\circ \).

### Formula to use:
For angles formed by a tangent and a secant intersecting outside a circle, the relationship is:

\[
\theta = \frac{1}{2} (\text{measure of the larger intercepted arc} - \text{measure of the smaller intercepted arc})
\]

In this case, the larger arc is \( (14x - 22)^\circ \), and the smaller arc would be \( 0^\circ \) because no second arc is mentioned (implying that the secant cuts off one arc only).

Thus, the equation becomes:

\[
23 = \frac{1}{2} (14x - 22)
\]

### Solving the equation:
First, multiply both sides by 2 to eliminate the fraction:

\[
46 = 14x - 22
\]

Next, add 22 to both sides:

\[
46 + 22 = 14x
\]

\[
68 = 14x
\]

Finally, divide both sides by 14:

\[
x = \frac{68}{14} = 4.857
\]

Thus, the value of \( x \) is approximately 4.857.

Would you like further clarification or details?

### Related Questions:
1. How does the relationship between tangents and secants apply in other circle problems?
2. What happens if the smaller intercepted arc has a nonzero value in such problems?
3. Can this concept be extended to other geometric figures beyond circles?
4. How do inscribed angles compare with tangent-secant angles in a circle?
5. What would the equation look like if a second secant was involved?

### Tip:
Always check the type of angle formed outside the circle—different cases (tangent-tangent, secant-secant) require slightly different formulas!

Solve for x based on the diagram of a circle, where the external angle formed by a tangent and secant is 23 degrees, and the arc subtended by the secant is (14x - 22) degrees.

Learn how to solve for x in a problem involving a circle with a tangent and secant intersecting outside the circle. The external angle is 23 degrees, and the arc subtended by the secant is (14x - 22) degrees. This problem applies the Tangent-Secant Angle Theorem.

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