Thales' Theorem this is how the mathematical property that relates the measurements of the straight segments formed by a bundle of parallel lines cut by straights transversals. Before talking about the theorem itself, it is good to remember the concept of a bundle of parallel lines, transverse lines and one of its properties:
two or more straight they are parallel when they have no common ground. When we highlight three or more parallel lines in a plane, we say that they form a beam in straightparallel. the straights transversals are those that “cut” the parallel lines.
Suppose a bundle of straightparallel form congruent line segments on a line cross any. In this hypothesis, it also forms congruent segments in any other transversal line.
The following image shows a bundle of straightparallel, two transversal lines and the measurements of the line segments formed by them.
Thales' Theorem
Line segments formed on straight lines transverse to a bundle of parallel lines are proportional.
This means that it is possible that divisions between the lengths of some segments formed under these circumstances will have the same result.
To better understand the stated theorem, look at the following image:
what the theorem in tales guarantees regarding the segments formed on the straighttransversals is the following equality:
JK = ON
KL NM
Note that the division was done, in this case, from top to bottom. You segments superior on the straights transversals appear in the numerator. O theorem it also guarantees other possibilities. Look:
KL = NM
JK ON
Other variations can be obtained by exchanging membership ratios or by applying the fundamental property of proportions (the product of means is equal to the product of extremes).
Other possibilities of proportionality by theorem of such are:
JK = KL
ON NM
ON = NM
JK KL
JK = ON
JL OM
KL = NM
JL OM
so much this theorem how much this property are used to find the measure of one of the segments when knowing the measure of the other three or when knowing the reasoninproportionality between two segments. The most important thing to solve exercises involving Thales' theorem is respect the order where line segments are placed in fractions.
Examples:
In the following bundle of parallel lines, we will determine the length of the NM segment.
Solution:
Let x be the length of the segment NM, let's show the proportionality between the segments and use the fundamental property of proportions to solve the equation:
2 = 4
8x
2x = 32
x = 32
2
x = 16 cm.
Note that 8 = 2·4 and that 16 is also equal to 2·4. This happens because, in the configuration used, the reasoninproportionality é 1/4. Also note that any of the reasons above could have been used to solve this problem and the result would be the same.
From the following image, let's calculate the JK segment measure.
Solution:
Let's choose one of the reasons described in theoremintales, replace the values given in the exercise and use the fundamental property of proportions, i.e:
4x - 20 = 20
6x + 30 = 40
40(4x – 20) = 20(6x + 30)
160x - 800 = 120x + 600
160x - 120x = 600 + 800
40x = 1400
x = 1400
40
x = 35
To find out the length of JK, we have to solve the following expression:
JK = 4x – 20
JK = 4·35 – 20
JK = 140 - 20
JK = 120
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-teorema-tales.htm