Problem 8.4.1. A rectangle was cut into six smaller rectangles, the areas of five of them are marked on the diagram. Find the area of the remaining rectangle.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ea0b100610baa73554bg-05.jpg?height=632&width=652&top_left_y=88&top_left_x=401)
Answer: 101.

Solution. Let's introduce the notation as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ea0b100610baa73554bg-05.jpg?height=652&width=701&top_left_y=901&top_left_x=380)

Notice that

$$
2=\frac{40}{20}=\frac{S_{H I L K}}{S_{D E I H}}=\frac{H K \cdot H I}{H D \cdot H I}=\frac{H K}{H D} \quad \text { and } \quad 2=\frac{126}{63}=\frac{S_{A B H G}}{S_{G H K J}}=\frac{B H \cdot G H}{H K \cdot G H}=\frac{B H}{H K}
$$

Therefore, if $H D=x$, then $H K=2 x$, and $B H=4 x$. From this, it follows immediately that $B D=3 x=D K$.

![](https://cdn.mathpix.com/cropped/2024_05_06_1ea0b100610baa73554bg-06.jpg?height=648&width=698&top_left_y=96&top_left_x=382)

We obtain that the rectangles $B C F D$ and $D F M K$ are equal, as are their areas. Thus,

$$
S_{E F M L}=S_{D F M K}-S_{D E I H}-S_{H I L K}=161-20-40=101
$$