Rectangle $A B C D$ has side $A B$ of length $12 \, \text{cm}$ and side $B C$ of length $6 \, \text{cm}$. Point $S$ is the center of the rectangle, and point $F$ lies on side $C D$. Quadrilateral $B C F S$ has one-third the area of rectangle $A B C D$.

Determine the length of segment $C F$.

(E. Novotná)

![](https://cdn.mathpix.com/cropped/2024_04_17_4bbdd946fc9507df08ddg-3.jpg?height=411&width=828&top_left_y=1825&top_left_x=614)

Note: The image is for illustration only.
The area of rectangle $ABCD$ is $72 \, \text{cm}^2 (12 \cdot 6 = 72)$, so quadrilateral $BCFS$ has an area of $24 \, \text{cm}^2 (72 : 3 = 24)$.

In the following considerations, we will use additional division determined by the midpoints of the sides of rectangle $ABCD$:

![](https://cdn.mathpix.com/cropped/2024_04_17_4bbdd946fc9507df08ddg-4.jpg?height=414&width=828&top_left_y=187&top_left_x=617)

The dotted lines divide the given rectangle into four identical rectangles with dimensions $6 \, \text{cm}$ and $3 \, \text{cm}$. The area of each of these rectangles is $18 \, \text{cm}^2 (18 = 72 : 4 = 6 \cdot 3)$. Quadrilateral $BCVS$ has an area of $27 \, \text{cm}^2 \left(18 + \frac{1}{2} \cdot 18 = 27\right)$. This is more than $24 \, \text{cm}^2$, so point $F$ actually lies on segment $VC$ (and not on segment $VD$).

For quadrilateral $BCFS$ to have the given area, we need to subtract $3 \, \text{cm}^2$ corresponding to triangle $FVS$. This triangle is right-angled and side $VS$ has a length of $3 \, \text{cm}$, so side $VF$ must have a length of $2 \, \text{cm} \left(\frac{1}{2} \cdot 3 \cdot 2 = 3\right)$. The length of segment $CF$ is $4 \, \text{cm} (6 - 2 = 4)$.

Evaluation. 2 points for the area of quadrilateral $BCFS$ and additional considerations; 2 points for the result; 2 points for the quality of the commentary.

Note. The previous considerations can be replaced by testing possibilities in a (sufficiently fine) grid. The figure looks, up to scale, like this:

![](https://cdn.mathpix.com/cropped/2024_04_17_4bbdd946fc9507df08ddg-4.jpg?height=457&width=822&top_left_y=1342&top_left_x=617)