1. **Problem statement:** We have a circle with center $O$ and points $A$, $B$, $C$, and $D$ on its circumference. A tangent line $FDE$ touches the circle at point $D$. We need to find the value of $y - x$, where $x$ is the angle at the center $O$ between points $B$ and $D$, and $y$ is the angle at the circumference at point $A$ between points $A$ and $D$. 2. **Key theorem:** The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at any point on the circumference. Mathematically, if $x$ is the central angle and $y$ is the inscribed angle subtending the same chord or arc, then: $$y = \frac{x}{2}$$ 3. **Tangent-secant angle property:** The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. Here, the tangent $FDE$ touches the circle at $D$, so the angle between the tangent and chord $BD$ equals the angle subtended by chord $BD$ at the circumference on the opposite side. 4. **Applying the properties:** Since $x$ is the angle at the center subtended by chord $BD$, and $y$ is the angle at the circumference subtended by the same chord $BD$, we have: $$y = \frac{x}{2}$$ 5. **Calculate $y - x$:** $$y - x = \frac{x}{2} - x = -\frac{x}{2}$$ 6. **Interpretation:** The value $y - x$ is negative half of $x$. Since angles are positive measures, the magnitude is $\frac{x}{2}$ but the expression $y - x$ equals $-\frac{x}{2}$. **Final answer:** $$y - x = -\frac{x}{2}$$