por We explain what the homologous sides are, with examples and solved exercises
What are homologous sides?
The homologous sides in two flat geometric figures are those that correspond to each other, keeping similarity. For example, the right hand of one person is homologous with the right hand of another person.
In plane geometry, there are not only homologous sides, but homologous vertices and angles as well. To see it, consider the following figure, which consists of two identical triangles ABC and A’B’C’:
When comparing them, it is clearly observed that the sides AB and A’B’ in blue are homologous, since they occupy a similar position in each triangle. The sides BC and B’C’ in purple are also homologous. And finally, the side AC in red color is homologous to the side A’C’.
Explanation
From the above, it follows that the homologous sides are those that occupy the same relative position in figures of the same shape. In the previous image, two identical triangles were used to show the idea, but it can be easily generalized to other flat geometric figures, formed by consecutive sides that close.
These figures are called polygons. For example, triangles and quadrilaterals are polygons with 3 and 4 sides, respectively.
The concept of homologous sides is important because it allows defining similarity criteria between polygons, as will be seen shortly. Similar figures have exactly the same shape and have the same proportion between their sides, even though they are not the same size.
And although until now reference was made only to flat figures, there are also similar figures in three dimensions. They are easily seen on supermarket shelves, when the same product is sold in identical containers, but with different sizes.
Other words that are used interchangeably in geometry to refer to homologous sides in geometric figures are: corresponding sides, respective sides, and equivalent sides.
Homologous vertices and angles
As with the sides, homologous vertices are also defined, which join pairs of homologous sides. For example, the vertices A and A’ of the previous figure are homologous. Similarly, the pairs of vertices B and B’ and C and C’ are homologous.
Finally, homologous angles occupy the same relative position in the figures. The vertices of the homologous angles are in turn homologous.
To illustrate the idea, take the angle between the blue and purple sides of the left triangle, which can be denoted as ∠ABC. This angle has its counterpart in the angle ∠A’B’C’, of the triangle on the right.
The vertex of this angle is B, which, as previously indicated, is homologous with B’, and the other two pairs of homologous angles of the triangles shown are:
∠BCA and ∠B’C’A’
∠CAB and ∠C’A’B’
similarity of polygons
For any two polygons to be similar, the following conditions must be met:
All pairs of homologous angles have the same measure.
Its pairs of homologous sides are proportional.
The two conditions must be met simultaneously to ensure similarity. You immediately see why.
In the following figure there are two quadrilaterals that are obviously not similar. This is because the first similarity condition is satisfied, but the second is not:
Although in the figures their pairs of homologous angles have the same measure, because they are all right angles (they measure 90º), the figures are not similar, because their pairs of sides are not proportional.
Instead, these two quadrilaterals have homologous sides with equal measure, but homologous angles do not measure equal. Therefore, the figures are clearly not similar.
similarity reason
If two figures are similar, the quotient between the homologous sides is the same and is called reason of similarity.
Denoting the sides of one of the figures as a, b, c, d… and the corresponding ones of the other figure as a’, b’, c’, d’…, the figures are similar if:
Perimeters and Areas of Similar Figures
The similarity ratio allows obtaining relationships between the perimeters, areas and volumes of two similar figures.
Ratio of perimeters of two similar figures
The perimeter P of a polygon is defined as the sum of all its sides. If you have a figure whose sides are a’, b’, c’, d’…, its perimeter P’ is:
P’ = a’ + b’ + c’ + d’ ….
If another polygon is similar to this one, and its sides are a, b, c, d…, it is true that:
And therefore:
a = r∙a’
The same can be said for the other sides of this figure. So the perimeter P is expressed as:
P = a + b + c + d…. = r∙a’ + r∙b’ + r∙c’ + r∙d’ + …
Since “r” is a factor common to all the addends, the relationship between P and P’ is:
P = r∙P’
This means that the ratio of the perimeters between two similar polygons is equal to the ratio of similarity.
Ratio of areas of two similar figures
If two similar figures have areas A and A’, respectively, they are related by:
A = r2∙A’
Where «r» is the ratio of similarity of the figures.
Ratio of volumes of two similar figures
Consider two similar three-dimensional figures, whose volumes are, respectively, V and V’. The relationship between them, through «r» is:
V = r3∙V’
examples
Blueprints
Portions of land, the plan of a building or even a piece of clothing can be represented at a smaller scale on a sheet of paper. Blueprints have the advantage of being easy to take with you and make the pertinent modifications before putting them into practice on the actual object.
maps
They are usually representations in the plan of a large area of land, from a village to the continents. They are also made to a certain scale.
They have numerous applications and there are many types. For example, by means of a map the terrain can be described, and when being located on a specific point, the best route to go from that point to another is determined.
models
They are three-dimensional scale representations of objects such as cars, buildings, and constructions in general.
solved exercise
The values that follow correspond to the sides of a pair of similar triangles. Find the ratio of similarity and the values of «x» and «y»:
triangle 1: 5, 8, 10
triangle 2:150,x,y
Solution
The ratio of similarity is the quotient:
r = 150/5 = 30
Therefore:
x = 30 × 8 = 240
y = 10 × 30 = 300