Problem 8.2. Given a rectangle $A B C D$. A line passing through vertex $A$ and point $K$ on side $B C$ divides the entire rectangle into two parts, the area of one of which is 5 times smaller than the area of the other. Find the length of segment $K C$, if $A D=60$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1281ceb10571c66f6e2bg-02.jpg?height=311&width=418&top_left_y=971&top_left_x=518)
Answer: 40.

![](https://cdn.mathpix.com/cropped/2024_05_06_1281ceb10571c66f6e2bg-02.jpg?height=313&width=422&top_left_y=1402&top_left_x=516)

Fig. 1: to the solution of problem 8.2

Solution. Draw a line through $K$ parallel to $AB$. Let it intersect side $AD$ at point $L$ (Fig. 1), then $ABKL$ and $DCKL$ are rectangles. Let $S_{ABK}=S$, then $S_{KLA}=S$. It is obvious that $S_{AKCD}=5S$ and $S_{LKCD}=5S-S=4S$.

The ratio of the areas of rectangles $ABKL$ and $DCKL$ with the common side $KL$ is equal to the ratio of the sides $BK$ and $CK$. Therefore, $\frac{BK}{CK}=\frac{2S}{4S}=\frac{1}{2}$, from which we get $KC=\frac{2}{3}BC=$ $\frac{2}{3} \cdot 60=40$