what is the value of x? x =

Explanation:

Step1: Recall central - inscribed angle relationship

The measure of a central angle is twice the measure of an inscribed angle subtended by the same arc. Here, the central angle $\angle SQR = 8x - 92^{\circ}$ and the inscribed angle $\angle STR=2x$. So, $8x - 92^{\circ}=2\times(2x)$.

Step2: Solve the equation

Expand the right - hand side: $8x - 92 = 4x$. Subtract $4x$ from both sides: $8x-4x - 92=4x - 4x$, which gives $4x-92 = 0$. Add 92 to both sides: $4x-92 + 92=0 + 92$, so $4x=92$. Divide both sides by 4: $x=\frac{92}{4}=23$.

Answer:

$23$