1. **State the problem:** We have a circle with center $Q$, points $P$ and $R$ on the circumference, and the angle $\angle PQR = 134^\circ$. Given $PQ = 7$ units, find the length of $PR$. 2. **Understand the geometry:** Since $Q$ is the center, $PQ$ and $QR$ are radii of the circle, so $PQ = QR = 7$ units. 3. **Use the Law of Cosines:** In triangle $PQR$, with sides $PQ = 7$, $QR = 7$, and angle $\angle PQR = 134^\circ$ between them, the length $PR$ can be found by: $$ PR^2 = PQ^2 + QR^2 - 2 \cdot PQ \cdot QR \cdot \cos(\angle PQR) $$ 4. **Substitute known values:** $$ PR^2 = 7^2 + 7^2 - 2 \cdot 7 \cdot 7 \cdot \cos(134^\circ) $$ 5. **Calculate:** $$ PR^2 = 49 + 49 - 98 \cdot \cos(134^\circ) $$ 6. **Evaluate $\cos(134^\circ)$:** Since $134^\circ = 180^\circ - 46^\circ$, $\cos(134^\circ) = -\cos(46^\circ) \approx -0.69465837$ 7. **Plug in the cosine value:** $$ PR^2 = 98 - 98 \cdot (-0.69465837) = 98 + 68.0886 = 166.0886 $$ 8. **Find $PR$ by taking the square root:** $$ PR = \sqrt{166.0886} \approx 12.89 $$ **Final answer:** The length of $PR$ is approximately **12.89** units.