Solved Problem on Flat Mirrors

A person of height H is in front of a vertical flat mirror. The distance from the eye of the observer to the ground is h, determine:
a) The lowest height d of this mirror so that the observer can see yourself with the full-body;
b) The distance r that the lower edge of the mirror is from the ground;
c) The height of the mirror and its floor distance depend on the distance from the observer to the mirror?

Problem data:

Height of the person (observer): H;
Distance from the eye from the observer to the floor: h.

Construction of the image:

We draw the \( \overline{AB} \), object, high H, at a distance x from the mirror (Figure 1).

Figure 1

On the other side of the mirror, we draw the image \( \overline{A'B'} \), with the same height H, with the same distance x from the mirror (Figure 2).

Figure 2

We draw a line from the point O, the eye of the observer, to the point A', head of the image, plot another line of the point O to point B', foot of the image (Figura 3).

Figure 3

From the segment \( \overline{OA'} \) intersecting with the mirror position, we obtain point C, and from the intersecting of the segment \( \overline{OB'} \), we have point D, so the segment \( \overline{CD} \) determines the size of the mirror (Figure 4).

Figure 4

Problem diagram:

Figure 5

Solution

a) To determine the size of the mirror, we will use the similarity of triangles. The triangle ΔOCD, height \( \overline{OG}=x \) and base \( \overline{CD}=d \) and the triangle ΔOA'B', height \( \overline{OO'}=2x \) and base \( \overline{A'B'}=H \)

\[ \begin{gather} \frac{\overline{CD}}{\overline{A'B'}}=\frac{\overline{OG}}{\overline{OO'}}\\ \frac{d}{H}=\frac{x}{2x} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {d=\frac{H}{2}} \]

b) To determine the distance of the lower edge from the mirror to the ground, segment \( \overline{DF}=r \) in the figure, we will use the similarity of triangles ΔB'DF with base \( \overline{{B'F}}=x \) and height \( \overline{DF}=r \), and the triangle ΔB'OB with base \( \overline{B'B}=2x \) and height \( \overline{OB}=h \)

\[ \begin{gather} \frac{\overline{DF}}{\overline{OB}}=\frac{\overline{B'F}}{\overline{B'B}}\\ \frac{r}{h}=\frac{x}{2x} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {r=\frac{h}{2}} \]

c) With the results obtained in items (a) and (b), we see that the size of the mirror and its height from the ground do not depend on the distance from the observer to the mirror. The size of the mirror (d) is directly proportional to the height of the observer (H), and the distance from the mirror to the ground (r) is directly proportional to the distance from the eyes of the observer to the ground (h).

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .