What is the reason for two segments

To understand Thales’ theorem you need to understand very well what is the reason between two segments.

For example, we have these two segments:

As you know, the segments are delimited by two extremes and are named by the extremes that limit it. The red segment, which begins at end A and ends at end B, is called segment AB.

If two segments have some relation between them, the same letters are used to name them, but since they cannot be repeated, the single quotation mark next to each letter is used and the quotation mark reads “prima”. So A’ would read “A prima”.

Thus, the blue segment, which starts at A’ and ends at B’, will be called A’ B’ segment.

Box]The ratio (or ratio) of two segments is the result of dividing the length of those two segments.[/box]

If the AB segment measures 5 cm and the A’ B’ segment measures 10 cm, what is the reason for these two segments?

All we have to do is divide the length of segment AB by the length of segment A’ B’:

The ratio of these two segments is 0.5, which means that AB is half that of A’ B’.

We can also calculate the ratio by dividing the length of segment A’ B’ by the length of segment AB:

In this case, the ratio is 2, or in other words, segment A’ B’ is twice as high as segment AB.

If you notice, to say that the AB segment is half as much as the A’ B’ segment, is the same as saying that the A’ B’ segment is twice as much as the AB segment.

Therefore, it is not necessary to calculate the ratio in both ways. Calculating it in one of two ways is enough.

Thales’ theorem

Once I’ve explained the reasoning between two segments and the proportionality between two pairs of segments, let’s look at Thales’ theorem.

We have two secant (non-parallel) straight lines. One is called the straight line r (red colour) and the other is called the straight line s (blue colour):

To these two straight lines, we cut them with several parallel lines (green colour), as follows:

To the points where they cut the lines parallel to the straight line, I will call them A, B and C and to the points where they cut the lines parallel to the straight line, I will call them A’, B’ and C’:

The green lines have divided the line r into two segments: segment AB and segment BC. We also have a third segment if we consider the first and last parallel line, i. e. the AC segment.

They have also divided the line s into two segments A’ B’ and B’ C’ and if we consider the first and last parallel line, there is a third segment A’ C’.

Thales’ theorem tells us the following:

Box]When any two straight lines, r and s, are cut by several parallel lines, the segments that make up the line r are proportional to the segments that make up the line s.[/box].

And what does that mean?

So if you divide the lengths of the segments that are at loggerheads, i. e. segment AB by segment A’ B’ they have the same reason as if you divide segment BC by segment B’ C’:

As they have the same reason, AB and A’ B’ are proportional to BC and B’ C’.

If we consider the segment formed by the first and last parallel line, i. e. the AC segment, it is also proportional to the AB segment:

And therefore, all segments of the line r are proportional to the segments of the line s:

What is Thales’ theorem for?

Thales’ theorem allows you to calculate the length of a segment, knowing the values of all the other segments of two straight lines that are in Thales’ position.

To be in Thales’ position means that the straight lines must be as the Thales theorem says, that is, two straight lines cut by several parallel straight lines.

We’ll solve several exercises to make it much clearer.