3. The perimeter of rectangle $A B C D$ is $66 \mathrm{~cm}$. Point $E$ is the midpoint of side $\overline{A B}$. The perimeter of triangle $\triangle E B C$ is $36 \mathrm{~cm}$, and the perimeter of quadrilateral ECDA is $60 \mathrm{~cm}$. What is the area of triangle $\triangle E B C$?
First method:

Sketch:

$1 \text{POINT}$

![](https://cdn.mathpix.com/cropped/2024_05_30_f1ff0db3c8ac61ddb2e9g-3.jpg?height=234&width=494&top_left_y=1708&top_left_x=290)

With the labels as shown in the figure, the perimeter of triangle $EBC$ is $x + y + z = 36 \text{ cm}$, the perimeter of quadrilateral $ECDA$ is $z + 2x + y + x = 60 \text{ cm}$, and the perimeter of rectangle $ABCD$ is $4x + 2y = 66 \text{ cm}$.

If we substitute $x + y + z = 36$ into the expression $z + 2x + y + x = 60$, we get $2x = 24 \text{ cm}, \text{i.e.,} \quad x = 12 \text{ cm}$.

From the condition $4x + 2y = 66$ and the fact that $x = 12$, we get $y = 9 \text{ cm}$.

The area of triangle $EBC$ is equal to half the product of the lengths of its legs, i.e., $p(EBC) = xy / 2$.

Substituting the values, we get $p(EBC) = 54 \text{ cm}^2$.