On a table with a circular top of diameter $0.6 \mathrm{~m}$, a square tablecloth with a side of $1 \mathrm{~m}$ is placed "crookedly". One corner of the tablecloth hangs over the edge of the table by $0.5 \mathrm{~m}$, and the adjacent corner by $0.3 \mathrm{~m}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_cd0342d3a611cc15345bg-4.jpg?height=531&width=512&top_left_y=2047&top_left_x=812)

Determine the length of the overhang of the remaining two corners.
Let's denote the vertices of the square tablecloth, which form the ends of two known corners, as $A$ and $B$; the remaining vertices of the square will be denoted as $C$ and $D$. The point $S$ indicates where the tablecloth touched the center of the table. The radius of the table is $0.3 \, \text{m}$.

![](https://cdn.mathpix.com/cropped/2024_04_17_cd0342d3a611cc15345bg-5.jpg?height=580&width=528&top_left_y=521&top_left_x=798)

In triangle $A B S$, we know from the problem statement all the sides: $|A B| = 1 \, \text{m}$, $|B S| = 0.6 \, \text{m}$, and $|S A| = 0.8 \, \text{m}$. Since $1^2 = 0.6^2 + 0.8^2$ (Pythagorean theorem), this is a right triangle.

Let $P$ be the foot of the altitude from $S$ to side $A B$. The area of this triangle can be expressed in two ways:

$$
S = \frac{1}{2} |B S| \cdot |S A| = \frac{1}{2} |A B| \cdot |P S|,
$$

from which we can derive the relationship for calculating the height $P S$:

$$
|P S| = \frac{|B S| \cdot |S A|}{|A B|} = \frac{0.6 \cdot 0.8}{1} = 0.48 \, \text{m}.
$$

Using the Pythagorean theorem, we can calculate the length of side $A P$ in triangle $A P S$ and then determine the length of segment $P B$:

$$
\begin{aligned}
& |A P| = \sqrt{|S A|^2 - |P S|^2} = \sqrt{0.8^2 - 0.48^2} = 0.64 \, \text{m}, \\
& |P B| = |A B| - |A P| = 1 - 0.64 = 0.36 \, \text{m}.
\end{aligned}
$$

In triangle $S C D$, let $Q$ be the foot of the altitude to side $C D$. The segments $D Q$ and $Q C$ correspond in length to segments $A P$ and $P B$. We can calculate the length of the height $S Q$:

$$
|S Q| = |P Q| - |P S| = 1 - 0.48 = 0.52 \, \text{m}.
$$

Using the Pythagorean theorem, we can calculate the lengths of the hypotenuses of triangles $S C Q$ and $S Q D$:

$$
\begin{aligned}
& |S C| = \sqrt{|Q C|^2 + |Q S|^2} = \sqrt{0.36^2 + 0.52^2} = \sqrt{0.4} \approx 0.63 \, \text{m}, \\
& |S D| = \sqrt{|Q D|^2 + |Q S|^2} = \sqrt{0.64^2 + 0.52^2} = \sqrt{0.68} \approx 0.82 \, \text{m}.
\end{aligned}
$$

When we subtract the radius of the table from these lengths, we find that the lengths of the overhangs of the remaining corners of the tablecloth are approximately $0.33 \, \text{m}$ and $0.52 \, \text{m}$.