## Task 3 - 280723

Let $A B C D$ be a trapezoid with $A B \| C D$, which satisfies the following conditions:

(1) The lengths of the sides $A B$ and $C D$ are in the ratio $5: 4$.

(2) The midline of the trapezoid has a length of $5.4 \mathrm{~cm}$.

(3) The height of the trapezoid is half the length of the side $A B$.

Determine whether the area of the trapezoid $A B C D$ is uniquely determined by these conditions. If so, give the area of the trapezoid in square centimeters!
Due to (1), $A B: C D=5: 4$, so $C D=\frac{4}{5} A B$. (4)

Since the length of the midline in a trapezoid is equal to half the sum of the lengths of the two parallel sides, due to (2) $\frac{1}{2}(A B+C D)=5.4 \mathrm{~cm}$. From this and (4) it follows that

$$
\frac{1}{2}\left(A B+\frac{4}{5} A B\right)=5.4 \mathrm{~cm} \Rightarrow A B=6 \mathrm{~cm}
$$

From (3) it follows that: The height of the trapezoid is $3 \mathrm{~cm}$. From this and (2), using the formula for the area of the trapezoid, it follows that (since $5.4 \cdot 3=16.2$) the area is uniquely determined by the given conditions (1) to (3); it is $16.2 \mathrm{~cm}^{2}$.