## 4. task

Let $A B C D$ be a square with side lengths of $31 \, \text{cm}$. On the side $\overline{A B}$, a point $E$ is chosen such that $|A E| = 11 \, \text{cm}$, on the side $\overline{B C}$, a point $F$ is chosen such that $|B F| = 14 \, \text{cm}$, and on the side $\overline{C D}$, a point $G$ is chosen such that $|C G| = 10 \, \text{cm}$. Determine the area of triangle $E F G$.
## Solution:

The area of triangle EFG is obtained by subtracting the area of rectangle AEHD and the areas of three right-angled triangles EGH, EBF, and FCG from the area of rectangle ABCD:

$$
P_{E F G}=P_{A B C D}-\left(P_{A E H D}+P_{E G H}+P_{G F C}+P_{E B F}\right)
$$

Since the area of a right-angled triangle is half the area of the rectangle whose sides correspond to the legs of the triangle that lie along the right angle, it follows:

$P_{E F G}=31 \cdot 31-(11 \cdot 31+10 \cdot 31: 2+10 \cdot 17: 2+20 \cdot 14: 2)$

$P_{E F G}=961-(341+155+85+140)=961-721=240$

Therefore, the area of triangle EFG is $240 \mathrm{~cm}^{2}$

![](https://cdn.mathpix.com/cropped/2024_05_30_8f249fc2b96bc1b5a03dg-4.jpg?height=647&width=683&top_left_y=533&top_left_x=1339)