From an earlier post "Laser distance meter and Pythagorean theorem" we can read, how easy it is to measure the heights of different objects if we can use a laser distance meter. By pressing some buttons, the laser distance meter performs the calculations and the height of the object is displayed on the screen.

However, in this post we will look at how to estimate the heights of objects, using only a mirror. In fact, it can be any reflective surface, even a smartphone can sometimes be used.

In the first example, we look at how to estimate the height of an object if we know the distance to the object. In the second example, we look at a situation where the distance to the object being measured is unknown.

Let's say you need to determine the height of a tree growing in your garden. At a certain distance from the three, a mirror must be placed on flat ground. Move away from the mirror, so that only the top of the tree is visible on the mirror. We know from the law of reflection that if the light hits a mirror at a certain angle, it will be reflected at the same angle. Thus, two similar triangles are formed. Knowing the properties of similar triangles, we can easily find the height of the tree.

Lengths AC and CD are easily measurable. Height DE is the height of the eyes from the ground. The triangles ABC and DEC are similar because their angles are equal. It follows from the similarity of the triangles, that AB : DE = AC : DC. So we get the height of the three: AB =(AC)(DE)/DC.

Let's calculate the height of the tree, if the distance to the tree is 12 meters, the distance to the mirror is 2 meters, and the height of the eyes from the ground is 1.5 meters:

But now suppose, we want to determine the height of a tree behind some kind of obstacle and we cannot measure the distance to that tree. It turns out, that we can measure the height of a tree, using the same method, but applying it twice.

We place the mirror at some distance from the tree as in the first example and move away from it until the top of the tree is visible in the mirror. Then we measure the length CD. Now we can place the mirror at the point where we just stood (point D) and move away from that point until we see the top of the tree in the mirror again. The length of DF is also easily measurable. The heights DE and FG are equal (DE = FG)

Now we have two pairs of similar triangles and we can write: AB : DE = AC : CD and AB : FG = (AC + CD) : DF. Height AB and length AC are unknown. The height of the tree AB can be obtained as follows:

Thus, the height of the tree is obtained by multiplying the height of the eyes from the ground by the distance between the positions of the mirror and dividing it by the difference between the distances between the observer and the mirror.

For example, let's say, that the height of the eyes from the ground is 1.5 m, the distance between the positions of the mirror is 2.5 m, and the difference between the distances between the observer and the mirror is 0.5 m.

As you can see, if you happen to have a mirror in your pocket, it is possible to measure the heights of objects without using a laser distance meter.

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