1. **State the problem:** We have kite quadrilateral $WXYZ$ with diagonals intersecting at $U$. Given lengths are $WU=12$, $UY=12$, $XU=6$, and $UZ=24$. We need to find the length $YX$. 2. **Recall kite properties:** In a kite, the diagonals are perpendicular, and one diagonal is bisected by the other. Here, $WU=UY=12$ means diagonal $WY=24$ is bisected at $U$. The other diagonal $XZ$ has segments $XU=6$ and $UZ=24$, so $XZ=30$. 3. **Use the Pythagorean theorem:** Since diagonals are perpendicular, triangles $WUX$ and $YUZ$ are right triangles. To find $YX$, consider triangle $XYU$. 4. **Find $YX$ using triangle $XYU$:** Points $X$ and $Y$ connect through $U$. The length $YX$ can be found by the Pythagorean theorem: $$YX=\sqrt{(XU)^2 + (UY)^2} = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180}$$ 5. **Simplify $\sqrt{180}$:** $$\sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$$ 6. **Final answer:** $$\boxed{YX = 6\sqrt{5}}$$