1. **State the problem:** We have a square window with area 81 cm², so each side is $\sqrt{81} = 9$ cm. The square is divided into 6 triangles of equal area by lines from three points on the bottom side. A fly sits at the point where all six triangles meet. We need to find the distance from the bottom of the window to the fly. 2. **Understand the setup:** The square has side length 9 cm. The bottom side is divided into 3 equal segments of length 3 cm each (since 9/3=3). Lines are drawn from these points to the top corners and to the opposite side, creating 6 triangles of equal area. 3. **Key fact:** All 6 triangles have equal area, so each triangle has area $\frac{81}{6} = 13.5$ cm². 4. **Find the coordinates:** Place the square in the coordinate plane with bottom-left corner at $(0,0)$ and top-right at $(9,9)$. The bottom points dividing the base are at $(3,0)$ and $(6,0)$. 5. **Lines from bottom points to top corners:** From $(3,0)$ to $(0,9)$ and $(9,9)$. From $(6,0)$ to $(0,9)$ and $(9,9)$. 6. **The six triangles meet at a single point inside the square, call it $P=(x,y)$. This point lies on the three lines connecting the bottom division points to the top corners. 7. **Find equations of lines:** - Line from $(3,0)$ to $(0,9)$: slope $m_1=\frac{9-0}{0-3}=-3$, equation $y=-3(x-3)$ or $y=-3x+9$. - Line from $(3,0)$ to $(9,9)$: slope $m_2=\frac{9-0}{9-3}=\frac{9}{6}=1.5$, equation $y=1.5(x-3)$ or $y=1.5x-4.5$. - Line from $(6,0)$ to $(0,9)$: slope $m_3=\frac{9-0}{0-6}=-1.5$, equation $y=-1.5(x-6)$ or $y=-1.5x+9$. - Line from $(6,0)$ to $(9,9)$: slope $m_4=\frac{9-0}{9-6}=3$, equation $y=3(x-6)$ or $y=3x-18$. 8. **The point $P$ lies on one line from each bottom point, so it lies on one line from $(3,0)$ and one from $(6,0)$. Try intersection of $y=-3x+9$ and $y=3x-18$: Set $-3x+9=3x-18$ gives $6x=27$ so $x=4.5$, then $y=-3(4.5)+9=-13.5+9=-4.5$ (outside square, discard). Try intersection of $y=-3x+9$ and $y=-1.5x+9$: Set $-3x+9=-1.5x+9$ gives $-3x=-1.5x$ or $-1.5x=0$ so $x=0$, $y=9$ (top left corner, no). Try intersection of $y=1.5x-4.5$ and $y=-1.5x+9$: Set $1.5x-4.5=-1.5x+9$ gives $3x=13.5$ so $x=4.5$, $y=1.5(4.5)-4.5=6.75-4.5=2.25$. Try intersection of $y=1.5x-4.5$ and $y=3x-18$: Set $1.5x-4.5=3x-18$ gives $-1.5x=-13.5$ so $x=9$, $y=1.5(9)-4.5=13.5-4.5=9$ (top right corner, no). 9. **So $P=(4.5,2.25)$ is the intersection inside the square where the six triangles meet. 10. **Distance from bottom:** The fly is at height $y=2.25$ cm from the bottom. **Final answer:** The fly is sitting 2.25 cm from the bottom of the window.