To solve for $x$ in this diagram, we can use the Pythagorean theorem since we have a right triangle within the circle. The radius $x$ serves as the hypotenuse of the triangle, with one leg of 6 and the other of $26 - x$.

Using the Pythagorean theorem:

$x^2 = 6^2 + (26 - x)^2$

1. Simplify each term:

   • $6^2 = 36$
   • $(26 - x)^2 = 676 - 52x + x^2$

2. Substitute these into the equation: $x^2 = 36 + 676 - 52x + x^2$

3. Cancel $x^2$ from both sides: $0 = 712 - 52x$

4. Solve for $x$: $52x = 712$ $x = \frac{712}{52} = 13.7$

Thus, the value of $x$ is approximately 13.7. Rounding to the nearest tenth, we match this with option B. 14.3.

Answer:

B. 14.3

Would you like further details or have any questions?

Here are five related questions:

1. How would the solution change if the radius $x$ was not the hypotenuse?
2. What properties of circles help determine the structure of this problem?
3. Could this problem be solved using trigonometric identities?
4. How does the Pythagorean theorem apply in circles beyond this example?
5. If the triangle were not right-angled, what alternative methods could solve for $x$?

Tip: Remember to check that the components of a right triangle satisfy the Pythagorean theorem when dealing with geometric problems involving circles and radii.