The Geometry became a part of Mathematics that satisfied me as a teenage girl who filled her pencil case with colors and highlighters, I liked having that set of squares from the LION brand, because it came with its plastic case green in color and the edges were smooth. A true artist to make the lines, segments, triangles and other geometric figures that we drew to learn about reasons
and proportions
between 2 line segments.

When we draw a straight line, we can indicate its length in any unit of measurement and we will refer to that continuous line of points as an unambiguous line. To use a nomenclature accepted throughout most of the world, we will write a lowercase r
to refer to the line and we will include the values for the endpoints of the line as P
•────• Q

When we try to divide that line into work sections, then we will refer to a line segment by placing a vinculum or a small dash over the pair of lowercase letters that we have assigned to the line segment, for example (

)

I am going to use the Geogebra graphic tool to explain this topic of the ratio and proportion of 2 line segments. Let the 2 segments be: ab = 5 cm cd = 7 cm

Note that I do not include the link above the letters of each segment because this editing format does not allow it. The RATIO is the division between the lengths of 2 line segments, it is frequently expressed in terms of fractions. The first segment (ab) corresponds to 5 cm of the 7 cm of the second segment (cd), usually the larger number is placed in the denominator.

We can also continue to expand the study of other segments and present a fraction that gives us an account of the ratio between this new pair of segments: ef = 5/2 cm gh = 7/2 cm

The ratio between these two segments, evidently smaller than segments (ab) and (cd), turn out to have the same relationship,

Ratio = 5/7

.

Since the analysis of each pair of segments separately gives us the result of the length ratio

Ratio = 5/7

, then we are in the presence of an equality of proportions according to the ratio.

The first 2 segments are proportional to the last 2 segments studied, because they have the same length ratio.

This same principle of the ratio between a pair of segments can be expanded and applied to some geometric figures, such as: square, rectangle, triangle, etc.

^{https://picasion.com/}

We draw a square of 2 cm on each side, AB = 2 cm, BC = 2 cm, CD = 2 cm and DA = 2 cm, then we set any coordinates (in this case (-1, -1)) of the center or focus for the proportional enlargement or reduction of this geometric figure taking as an example what was done with the ratio and proportion of straight line segments.

In this case it is called

Homothecy Ratio

(k), where each side of the original figure is going to be multiplied by this value (k), resulting in the geometric figure of a square increased by a value of 2, which was the homothecy ratio that I imposed on the example.

Finally, I present the case of proportionality by the homothecy ratio of a triangle with sides: AB = 2 cm, BC = 2 cm and CA = 2.8 cm.

Bibliographic support and image source

Our ideas and knowledge on the subject discussed in this article can be expanded voluntarily by consulting the following catalogue of references:

Geogebra Calculator: Linear function cover
Wikipedia: Vinculum
Blog: Straight alphabet
Wikipedia: Segment
Virtual program: Geogebra
Blog: The proportions
Blog: Ratios and proportions

Graphical representation,
remains a powerful tool
when teaching Mathematics classes