1. **Problem statement:** Find the length of segment $AF$ on trapezoid $DEFG$ where $\angle DEF = 90^\circ$, bases $DE = 6$ and $EF = 30$ are parallel, height $DE = 20$, and point $A$ lies on $FG$ such that the area of triangle $\triangle AEF$ equals the area of quadrilateral $AEDG$. 2. **Step 1: Understand the trapezoid and given data.** - $DEFG$ is a trapezoid with bases $DE = 6$ and $EF = 30$ parallel. - $\angle DEF = 90^\circ$ means $DE$ is vertical and $EF$ is horizontal. - Height $DE = 20$ means the vertical distance between bases is 20. 3. **Step 2: Assign coordinates for clarity.** - Let $D = (0,0)$. - Since $DE = 6$ vertical, $E = (0,6)$. - $EF = 30$ horizontal, so $F = (30,6)$. - Since height is 20, $G$ lies horizontally aligned with $D$ at $y=0$, so $G = (30,0)$. 4. **Step 3: Coordinates summary:** - $D(0,0)$, $E(0,6)$, $F(30,6)$, $G(30,0)$. - $FG$ is segment from $F(30,6)$ to $G(30,0)$. 5. **Step 4: Point $A$ lies on $FG$, so $A = (30,y)$ for some $y$ between 0 and 6.** 6. **Step 5: Calculate areas:** - Area of trapezoid $DEFG$ is $\frac{(DE + FG)}{2} \times$ height. - Since $FG$ is vertical segment from $(30,6)$ to $(30,0)$, length $FG = 6$. - Height between bases is horizontal distance $DE$ to $FG$, which is 30 units. - So area $= \frac{(6 + 6)}{2} \times 30 = 6 \times 30 = 180$. 7. **Step 6: Area of quadrilateral $AEDG$ is total trapezoid area minus area of triangle $AEF$.** 8. **Step 7: Area of triangle $AEF$ with vertices $A(30,y)$, $E(0,6)$, $F(30,6)$:** - Base $EF$ length = 30. - Height is vertical distance from $A$ to line $EF$ at $y=6$, so height $= |6 - y|$. - Area $\triangle AEF = \frac{1}{2} \times 30 \times |6 - y| = 15 |6 - y|$. 9. **Step 8: Given area $\triangle AEF = $ area quadrilateral $AEDG$, so:** $$15 |6 - y| = 180 - 15 |6 - y|$$ 10. **Step 9: Solve for $y$:** $$15 |6 - y| + 15 |6 - y| = 180$$ $$30 |6 - y| = 180$$ $$|6 - y| = 6$$ 11. **Step 10: Since $y$ is between 0 and 6, $|6 - y| = 6 - y$, so:** $$6 - y = 6 \implies y = 0$$ 12. **Step 11: Point $A$ is at $(30,0)$, which coincides with $G$.** 13. **Step 12: Length $AF$ is distance between $A(30,0)$ and $F(30,6)$:** $$AF = |6 - 0| = 6$$ **Final answer:** $$\boxed{6}$$