It is known that the length of the segment representing the product of two segments depends on the choice of the unit segment. How does the length of the product segment change if we reduce the unit segment to half of its original length?
Let the two given segments be $a$ and $b$. Denote the unit segment by $e$.

![](https://cdn.mathpix.com/cropped/2024_05_02_93a6e15d2af204a746f1g-1.jpg?height=353&width=331&top_left_y=222&top_left_x=886)

From the theorem of parallel chords, with the notation of the figure, we have

$$
\frac{e}{b}=\frac{a}{d}
$$

and from this

$$
d=\frac{a \cdot b}{e}
$$

Since $e$ is a unit length, $d$ is equal to the product segment.

It is visible: if we reduce the unit segment to half, the length of the product segment doubles. (If, however, we choose the length unit to be twice as large, then the product (linear) image decreases to half.)

Csorba István (Győr, Apáczai Csere J. Gyak. Ált. Isk., 8. o. t)