Two circles $\mathrm{s}$ are externally tangent. Segment $A B$ is the diameter of one circle and has a length of $6 \mathrm{~cm}$, segment $C D$ is the diameter of the other circle and has a length of $14 \mathrm{~cm}$. Quadrilateral $A B C D$ is a trapezoid with bases $A B$ and $C D$.

Determine the maximum area that trapezoid $A B C D$ can have, and explain why it cannot be larger.

(K. Pazourek)

![](https://cdn.mathpix.com/cropped/2024_04_17_5ebc08916757a9a5715ag-5.jpg?height=602&width=740&top_left_y=710&top_left_x=658)
The number of trapezoids $s$ with the given properties is infinite. All of them have the same base lengths, but their heights vary. However, this height cannot be greater than the distance between the centers of the circles:

![](https://cdn.mathpix.com/cropped/2024_04_17_5ebc08916757a9a5715ag-5.jpg?height=597&width=740&top_left_y=1683&top_left_x=658)

Let the centers of the circles be $O, P$ and the foot of the perpendicular from $O$ be $Q$. If points $P$ and $Q$ do not coincide, then the height $O Q$ is the leg of the right triangle $O Q P$, and therefore is smaller than its hypotenuse $O P$.

Thus, the trapezoid with the largest area is the one whose bases are perpendicular to the line segment connecting the centers of the circles:

![](https://cdn.mathpix.com/cropped/2024_04_17_5ebc08916757a9a5715ag-6.jpg?height=623&width=759&top_left_y=428&top_left_x=654)

The lengths of the bases of the trapezoid are $|A B|=6 \mathrm{~cm}$ and $|C D|=14 \mathrm{~cm}$, and the height is $|O P|=\frac{1}{2}(6+14)=10 \mathrm{~cm}$. The area of this trapezoid, and thus the maximum possible area, is

$$
\frac{1}{2}(|A B|+|C D|) \cdot|O P|=\frac{1}{2}(6+14) \cdot 10=100\left(\mathrm{~cm}^{2}\right)
$$

Evaluation. 2 points for recognizing that the bases remain constant while only the height changes; 2 points for the area of the largest trapezoid; 2 points for the justification and quality of the commentary.

Note. In the justification that segment $O P$ is greater than $O Q$, it is sufficient to argue that in a triangle, the side opposite the larger interior angle is larger.