Solution

Statement

$13.1.7.$ A girl who has forgotten geometric optics is looking at her reflection in a small mirror. To see a larger part of her image at once, she brings the mirror close to her face and then moves it away. Explain to the girl how the size of the visible part of her face changes depending on the distance to the mirror.

Solution

The size of the visible part of the face does not depend on the distance to the mirror and is always exactly equal to twice the size of the mirror itself.

Proof:
Let the linear size of the mirror be $d$, and the distance from the girl's eyes to the mirror be $L$. According to the laws of geometric optics, a flat mirror produces an erect virtual image located at the same distance $L$ behind the plane of the mirror. Therefore, the total distance from the observer's eyes to the image of the face is $2L$.

The mirror acts as a limiting "window" of size $d$ located at a distance $L$ from the eyes. Using the property of similar triangles with their vertex at the observer's eye:
The ratio of the linear size of the visible part of the face ($D$) to the distance to the image ($2L$) is equal to the ratio of the size of the mirror ($d$) to the distance to it ($L$).
$$\frac{D}{2L} = \frac{d}{L} \implies D = 2d$$
As we can see, the variable $L$ (distance to the mirror) cancels out. Consequently, a mirror of, say, 5 cm in size will always fit exactly 10 cm of the face, no matter how near or far it is held.

Why does the girl think otherwise?
When the distance changes, the angular size of the mirror and, consequently, the image seen in it changes. The closer the mirror is to the face, the larger the viewing angle it covers. Facial details become visually larger (taking up more space on the retina), which creates a psychological illusion that the visible area has narrowed. Conversely, if you move the mirror away, the angular size decreases, the details become smaller, and it seems that you see "more", even though the physical size of the reflected area remains exactly the same ($2d$).

P.S. We have rigorously proven everything and figured out where this optical illusion comes from. But let's re-read the problem statement carefully: *"Explain to the girl"*. Now try explaining this on your fingers, without similar triangles, virtual images, or laws of reflection, to a person who has completely forgotten physics but strongly believes their own eyes! I can't imagine how difficult it will be for whoever takes on this mission. So here is your creative homework with an asterisk: figure out how to explain this to the girl in real life so she actually believes you!
Good luuuck!!