To determine the side length of the smallest square plate on which a 26 cm chopstick can fit along the diagonal without any overhang, we need to use some basic concepts from geometry.
When a square has a diagonal of a certain length, we can relate the side length of the square to its diagonal using the Pythagorean theorem. Specifically, if the side length of the square is [tex]\( s \)[/tex] and the diagonal length is [tex]\( d \)[/tex], we have:
[tex]\[ d = s \sqrt{2} \][/tex]
In this problem, the diagonal [tex]\( d \)[/tex] is equal to 26 cm. We need to find the side length [tex]\( s \)[/tex] of the square that will allow this chopstick to fit perfectly across the diagonal.
We can rearrange the formula to solve for [tex]\( s \)[/tex]:
[tex]\[ s = \frac{d}{\sqrt{2}} \][/tex]
Now substitute [tex]\( d = 26 \)[/tex] cm:
[tex]\[ s = \frac{26}{\sqrt{2}} \][/tex]
To simplify this, we can multiply both the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ s = \frac{26 \sqrt{2}}{2} \][/tex]
[tex]\[ s = \frac{26 \cdot 1.414}{2} \][/tex] (since [tex]\( \sqrt{2} \approx 1.414 \)[/tex])
[tex]\[ s = \frac{36.764}{2} \][/tex]
[tex]\[ s = 18.382 \][/tex]
Finally, we round this result to the nearest tenth:
[tex]\[ s \approx 18.4 \][/tex]
Therefore, the side length of the smallest square plate on which a 26 cm chopstick can fit along the diagonal without any overhang is approximately 18.4 cm.
