Deg Mirror

In which hand they hold their machine guns?

The statue in front of corporate headquarters of "Wealth and
Health" insurance company depicts its founder Fat Tony with
Thompson machine gun in his right hand. The bulding is
V-shaped skyscraper with angle 120 deg between the
arms of 'V', and mirror glass windows.

In which hand the mirror images of Fat Tony hold their Thompsons?

Each reflection flips left and right, so a double reflection restores the parity of the image. If you have an even number of reflections, he holds it in his right hand, and if you have an odd number, he holds it in his left.

Light rays coming straight into the v only reflect once. They hit one side of the v and come out parallel to the other side (with a parity flip). Light rays coming in from an angle could get reflected twice (with no parity flip).

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180 deg band- man in the mirror

How do I calculate the angle of incidence?

1) A light ray travelling parallel to the optical axis in air (n=1) is incident on a spherical surface at a height of 0.02m. If the material to the right of the the surface has a refractive index of 1.6 and the radius of curvature of the surface is 0.05m, calculate the angle of incidence and angle of refraction. Answer i=21.8 deg, i'=13.42 deg

2) What is the length of the smallest mirror in which a man 180cm tall can see himself at full length? Answer 90cm

Please show the working to both questions!

1.
Let:
i be the angle of incidence,
r be the angle of refraction,
u be the refractive index,
O be the centre of curvature,
AP be the incident ray, meeting the sphere at P,
NO be the axis,
S be a point on radius OP produced,
T be the point on NO vertically below P.

AP || NO, and SPO is a transversal.
Corresponding angles APS (angle of incidence) and NOP are equal.

tan(i) = TP / TO = 0.02 / 0.05
i = 21,80 deg.

sin(i) / sin(r) = u
sin(r) = sin(i) / u
= sin(21.8 deg) / 1.6
r = 13.42 deg.

2.
Let:
E be the man's eyes,
P be his head in the image,
Q be his feet in the image.
M be the top of the mirror,
N be the bottom of the mirror.

The man can just see his entire image when EMP and ENQ are straight lines.

The image is as far behind the mirror as his eyes are in front.

Therefore MN is a midsegment line of triangle EPQ, and its length is half that of PQ.

The mirror has to be positioned at the correct height, if the man is to see his entire image.