Problem 3. Let $\triangle A B C$ with $m(\measuredangle A)=90^{\circ}$. Construct the square $B D E C$ in the half-plane delimited by $B C$ that does not contain $A$. The angle bisector of $\angle A$ intersects the sides $[B C]$ and $[D E]$ at $F$ and $G$ respectively. If $|A B|=16 \mathrm{~cm}$ and $|A C|=4 \mathrm{~cm}$, calculate the area of the quadrilateral $B D G F$.
Solution:

Drawing

$1 p$

![](https://cdn.mathpix.com/cropped/2024_06_07_04bd8e18437a78c4edccg-2.jpg?height=415&width=420&top_left_y=2275&top_left_x=1514)

$$
\begin{aligned}
& \Delta A B C \equiv \triangle B M D([B C] \equiv[B D], \measuredangle A B C \equiv \measuredangle B D M) \Rightarrow[A C] \equiv[B M] \text { (1) ............................................. } 1 \text{p}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_04bd8e18437a78c4edccg-3.jpg?height=68&width=1778&top_left_y=161&top_left_x=195)

$$
\begin{aligned}
& \text { From (1) and (2) } \Rightarrow[A C] \equiv[N D] \text {................................................................................................................................ }
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_04bd8e18437a78c4edccg-3.jpg?height=65&width=1784&top_left_y=298&top_left_x=192)

![](https://cdn.mathpix.com/cropped/2024_06_07_04bd8e18437a78c4edccg-3.jpg?height=107&width=1781&top_left_y=364&top_left_x=194)

$$
\begin{aligned}
& A_{B D G F}=A_{A M N}-2 \cdot A_{A B C} \\
& A_{B D G F}=200-64 \Rightarrow A_{B D G F}=136 \text{ cm}^{2}
\end{aligned}
$$