7. (10 points) On the hypotenuse $A B$ of a right triangle $A B C$, points $X$ and $Y$ are taken such that $A X=A C$ and $B Y=B C$. It turns out that $X Y=p$. Find the product $A Y \cdot B X$.

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# Solution:

![](https://cdn.mathpix.com/cropped/2024_05_06_9c64a952bd96df941010g-07.jpg?height=502&width=814&top_left_y=366&top_left_x=655)

By the triangle inequality $A C+C B>A B$, so the placement of points $X$ and $Y$ on the hypotenuse will be exactly as shown (see figure).

Let $A Y=a$ and $B X=b$, then $A C=a+p$ and $B C=b+p$. Using the Pythagorean theorem

$$
A C^{2}+B C^{2}=A B^{2} \Longrightarrow(a+p)^{2}+(b+p)^{2}=(a+b+p)^{2}
$$

Expanding the brackets and combining like terms

$$
2 a b=p^{2} \Longrightarrow a b=\frac{p^{2}}{2}
$$

## Answers:

| Option | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | 4.5 | 12.5 | 6.125 | 10.125 | 15.125 | 24.5 | 21.125 | 28.125 | 40.5 | 45.125 | 40.5 |