1. **Problem Statement:** Calculate the value of $x$ to 4 significant figures given two circles each of radius 10.5 cm with centers $A$ and $B$ touching at $T$. Given angles $\angle XAD = \angle YBC = 160^\circ$ and lines $XY$, $ATB$, and $DC$ are parallel. 2. **Understanding the Geometry:** - Each circle has radius $r = 10.5$ cm. - The circles touch at point $T$, so $AB = 2r = 21$ cm. - Angles $\angle XAD$ and $\angle YBC$ are $160^\circ$, which helps determine the positions of points $X$, $D$, $Y$, and $C$. - Lines $XY$, $ATB$, and $DC$ are parallel. 3. **Finding $x$:** - Since $x$ is a length related to the figure, and the problem states "x cm 10 cm Determine the value of x to 4 significant figures," we interpret $x$ as a segment length related to the geometry. 4. **Using the angle $160^\circ$ at $A$ and $B$:** - The angle at $A$ between points $X$ and $D$ is $160^\circ$. - The minor arc $AXD$ corresponds to this angle. 5. **Calculate the chord length $XD$ in the first circle:** - The chord length $XD$ subtending an angle $160^\circ$ at the center $A$ is given by: $$ XD = 2r \sin\left(\frac{160^\circ}{2}\right) = 2 \times 10.5 \times \sin(80^\circ) $$ - Calculate $\sin(80^\circ)$: $$ \sin(80^\circ) \approx 0.9848 $$ - Therefore: $$ XD = 21 \times 0.9848 = 20.681 $$ 6. **Calculate $x$:** - Since $x$ is likely the length $AX$ or a related segment, and given the problem context, assume $x = XD$ or a segment proportional to it. - The problem states "x cm 10 cm" which suggests $x$ is near 10 cm. - Using the geometry and parallel lines, $x$ corresponds to the length of segment $XY$ or a related segment. - Without additional numeric data, the best estimate for $x$ based on the chord length is: $$ x = 20.68 \text{ cm (rounded to 4 significant figures)} $$ **Final answer:** $$ x = 20.68 $$ cm to 4 significant figures.