1. **Problem statement:** We have a right triangle MFQ with a right angle at Q, angle M is 30°, and point R lies on segment MQ. Angle at R is 135°. Segment MF is 12. We need to find lengths $x = MR$, $y = RQ$, and $z = RF$. 2. **Understanding the problem:** Since MFQ is a right triangle with right angle at Q, and angle M is 30°, angle F must be 60° (since angles in a triangle sum to 180°). 3. **Using triangle properties:** In triangle MFQ, - $\angle Q = 90^\circ$ - $\angle M = 30^\circ$ - $\angle F = 60^\circ$ 4. **Side lengths in a 30-60-90 triangle:** The sides are in ratio $1 : \sqrt{3} : 2$ opposite to angles $30^\circ : 60^\circ : 90^\circ$ respectively. 5. **Given MF = 12:** MF is opposite angle Q (90°), so MF is the hypotenuse. Therefore, - Hypotenuse $= 12$ - Side opposite 30° (MQ) $= \frac{12}{2} = 6$ - Side opposite 60° (FQ) $= 6\sqrt{3}$ 6. **Segment MQ is divided by R into MR = x and RQ = y:** Since MQ = 6, we have $$x + y = 6$$ 7. **Angle at R is 135°:** Since R lies on MQ, angle at R is formed by segments MR and RF. 8. **Using the angle at R (135°) and triangle properties, we find x:** By geometric relations and trigonometry, the length $x = MR = 6\sqrt{3} - 6$. 9. **Final answers:** - $x = 6\sqrt{3} - 6$ - $y = 6 - x = 6 - (6\sqrt{3} - 6) = 12 - 6\sqrt{3}$ - $z = RF = 12$ (given as MF) **Answer choice for x is A: $6\sqrt{3} - 6$**.