1. **State the problem:** We have a quadrilateral with points A, P, Q, C, and B. Given lengths are AP = 6 cm, PQ = $x$, QC = 12 cm, PC = 18 cm, BC = 20 cm, and AB = $y$. We want to find relationships involving $x$ and $y$. 2. **Analyze the segments:** Note that PC = 18 cm is the segment from P to C, and since P lies between A and Q, and Q lies between P and C, the total length AC = AP + PQ + QC = 6 + $x$ + 12 = $18 + x$ cm. 3. **Check for possible right triangles or use the triangle inequality:** Since BC = 20 cm and AB = $y$, and points A, B, C form a triangle, we can use the triangle inequality or Pythagoras if right angles are known. 4. **Use the given lengths to find $y$ in terms of $x$ if possible:** Without additional angle information, we cannot directly solve for $x$ or $y$. However, if we assume the quadrilateral is a trapezoid or right-angled, we can apply the Pythagorean theorem. 5. **Assuming triangle ABC is right-angled at B:** Then, $$y^2 + 20^2 = (18 + x)^2$$ 6. **Expand and simplify:** $$y^2 + 400 = (18 + x)^2 = 18^2 + 2 \times 18 \times x + x^2 = 324 + 36x + x^2$$ 7. **Rearranged equation:** $$y^2 = 324 + 36x + x^2 - 400 = x^2 + 36x - 76$$ 8. **Final relationship:** $$y = \sqrt{x^2 + 36x - 76}$$ This expresses $y$ in terms of $x$ under the right-angle assumption at B. Without more information, this is the best relation we can provide.