1. **Problem statement:** We have a quadrilateral OXPY with diagonals OX and PY intersecting at point T. Given: - $OX = 3a$ - $OY = 12b$ - $YP = 5a - 2b$ We need to find the length $XY$ in terms of $a$ and $b$. 2. **Key property:** In a quadrilateral where diagonals intersect, the segments created by the intersection point satisfy the property: $$\frac{OT}{TX} = \frac{PT}{TY}$$ 3. Since $OX$ is a diagonal with length $3a$, and $OY$ is another diagonal with length $12b$, and $PY$ is given as $5a - 2b$, we can use the fact that the diagonals bisect each other proportionally. 4. Let $T$ divide $OX$ into segments $OT$ and $TX$, and $PY$ into $PT$ and $TY$. Since $T$ is the intersection of diagonals, the ratio of segments on one diagonal equals the ratio on the other: $$\frac{OT}{TX} = \frac{PT}{TY}$$ 5. We know $OX = OT + TX = 3a$ and $PY = PT + TY = 5a - 2b$. 6. Since $OY = 12b$ is vertical and $OX = 3a$ is horizontal, the segment $XY$ connects points $X$ and $Y$ with coordinates: - $X$ at $(3a,0)$ - $Y$ at $(0,12b)$ 7. The length $XY$ is the distance between points $X(3a,0)$ and $Y(0,12b)$: $$XY = \sqrt{(0 - 3a)^2 + (12b - 0)^2} = \sqrt{(3a)^2 + (12b)^2} = \sqrt{9a^2 + 144b^2}$$ **Final answer:** $$XY = \sqrt{9a^2 + 144b^2}$$