1. **Problem statement:** We have a kite-shaped quadrilateral WXYZ with diagonals WY and XZ intersecting at U. 2. **Given:** WY is divided into two segments 12 and 12 by U, so WY = 24. 3. XZ is divided into two segments 6 and 24 by U, so XZ = 30. 4. Sides WX and XY are equal, and sides WZ and ZY are equal. 5. To find the perimeter, we need the lengths of all four sides: WX, XY, WZ, and ZY. 6. Since WX = XY and WZ = ZY, perimeter = 2(WX + WZ). 7. Use the right triangles formed by the diagonals intersecting at U to find WX and WZ. 8. The diagonals of a kite are perpendicular, so triangles WUX and YUX are right triangles. 9. For WX (triangle WUX): - WU = 12 (half of WY) - XU = 6 (half of XZ) - WX = \sqrt{WU^2 + XU^2} = \sqrt{12^2 + 6^2} = \sqrt{144 + 36} = \sqrt{180} = 6\sqrt{5} 10. For WZ (triangle WUZ): - WU = 12 - ZU = 24 - WZ = \sqrt{WU^2 + ZU^2} = \sqrt{12^2 + 24^2} = \sqrt{144 + 576} = \sqrt{720} = 12\sqrt{5} 11. Calculate perimeter: $$\text{Perimeter} = 2(WX + WZ) = 2(6\sqrt{5} + 12\sqrt{5}) = 2(18\sqrt{5}) = 36\sqrt{5}$$ 12. **Final answer:** The perimeter of kite WXYZ is $36\sqrt{5}$ units.