# Ratio of distance between mirror and person

In perspective of a given example, if a man was to stand $2\ m$ away from a mirror which was $0.9\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be if the man was now $6\ m$ away from the mirror and was to maintain a full reflection? Would the mirror be equal to or less than the original height and why? This scenario seems to have caught me out multiple times. So, some reasoning would be much appreciated.

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You haven't said what the condition for the height the mirror would have to be is. Perhaps the requirement is that the man can see his full reflection? This is not an uncommon example problem in geometric optics and the answer is slightly surprising but very easy to get using the rules of reflection. –  dmckee Feb 15 at 20:47
Ah thank you for pointing out that critical fact! Post was edited to show that requirement –  Jared Ping Feb 15 at 20:49
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## 1 Answer

A way to find the solution is to

• Assume the mirror is very large.
• Draw the reflected rays from the top of his head to his eyes and from his toes to his eyes. Recall that angle of incidence equals angle of reflection and use geometry (similar triangles) to figure it out.
• Chop the mirror off just big enough to catch those rays.
• Use the geometric relationships to determine the dependence of the size on the distance he stands from the mirror.
• Smile when you see how simple the final answer is.
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Many thanks! From what I could establish, it would appear that the mirror would remain the same size due to the proportionality of the mirror becoming smaller in appearance as the man would increase his distance from it. Therefore still seeing his full reflection, just as a smaller image. –  Jared Ping Feb 15 at 21:14
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