1. **Problem statement:** We have a circle with center $O$ and a circumscribed angle $A$ around it. We want to find the length of segment $\overline{AB}$. Given are an angle of $73^{\circ}$ and lengths 4 and 3 (likely sides or segments related to the problem). 2. **Understanding the problem:** Since $A$ is circumscribed about circle $O$, $A$ is a tangent point or vertex of a polygon circumscribing the circle. The angle $73^{\circ}$ is likely an angle at $A$ or related to the polygon. 3. **Assumptions and approach:** Without a diagram, we assume $\overline{AB}$ is a side of the polygon touching the circle. The lengths 4 and 3 may be tangents from points $A$ and $B$ to the circle or sides adjacent to $AB$. 4. **Using the tangent-segment theorem:** Tangents from a point to a circle are equal in length. If $A$ and $B$ are points of tangency, then the tangents from $A$ and $B$ to the circle are equal. 5. **Calculate $\overline{AB}$:** If the angle at $A$ is $73^{\circ}$ and the adjacent sides are 4 and 3, then by the Law of Cosines in triangle $ABC$: $$\overline{AB}^2 = 4^2 + 3^2 - 2 \times 4 \times 3 \times \cos(73^{\circ})$$ 6. **Calculate the cosine value:** $$\cos(73^{\circ}) \approx 0.2924$$ 7. **Substitute and simplify:** $$\overline{AB}^2 = 16 + 9 - 24 \times 0.2924 = 25 - 7.0176 = 17.9824$$ 8. **Find $\overline{AB}$:** $$\overline{AB} = \sqrt{17.9824} \approx 4.24$$ **Final answer:** The length of $\overline{AB}$ is approximately $4.24$ units.