3. Let $ABCD$ be a rectangle whose side lengths, expressed in centimeters, are natural numbers. Point $P$ is the midpoint of side $\overline{BC}$, and point $B'$ is the reflection of point $B$ with respect to the line $AP$. If the perimeter of quadrilateral $ABPB'$ is $15 \, \text{cm}$, what is the minimum and maximum possible area of rectangle $ABCD$?
3. Let $ABCD$ be a rectangle whose side lengths, expressed in centimeters, are natural numbers. Point $P$ is the midpoint of side $\overline{BC}$, and point $B'$ is the axis-symmetric image of point $B$ with respect to the line $AP$. If the perimeter of quadrilateral $ABPB'$ is $15 \, \text{cm}$, what is the minimum and maximum possible area of rectangle $ABCD$?

## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-19.jpg?height=391&width=665&top_left_y=1415&top_left_x=224)

Let $|AB| = a$ and $|AD| = b$. Then, by the given axis symmetry, the length of $\overline{AB}$ is mapped to $\overline{AB'}$, and the length of $\overline{BP}$ is mapped to $\overline{PB'}$.

Thus, for the side lengths of quadrilateral $ABPB'$, we have $|AB| = |AB'| = a$ and $|BP| = |PB'| = \frac{b}{2}$. \quad 1 POINT

Now we have:

\[
\begin{aligned}
& o_{ABPB'} = 2 \cdot a + 2 \cdot \frac{b}{2} \\
& o_{ABPB'} = 2a + b \\
& 2a + b = 15
\end{aligned}
\]

Since $b = 15 - 2a$ and $a, b \in \mathbb{N}$, we have the following possibilities:

| $a$ | $b$ | $P_{ABCD}$ |
| :---: | :---: | :---: |
| 1 | 13 | $13 \, \text{cm}^2$ |
| 2 | 11 | $22 \, \text{cm}^2$ |
| 3 | 9 | $27 \, \text{cm}^2$ |
| 4 | 7 | $28 \, \text{cm}^2$ maximum possible area |
| 5 | 5 | $25 \, \text{cm}^2$ |
| 6 | 3 | $18 \, \text{cm}^2$ |
| 7 | $\overline{1}$ | $7 \, \text{cm}^2$ minimum possible area |

TOTAL 10 POINTS