Why are the squares both rhombuses and rectangles?

How many rectangles are there in the image? And how many rhombus? If you replied “nobody” then you were wrong! In the figure they are clearly visible 2 squares which, from a mathematical point of view, are at the same time both rectangles That rhombusall in the big family of parallelograms: quadrilaterals with the parallel sides two by two. It is a matter of rectal corners and of congruent sideswe explain why the square is considered a particular case.

The squares are rectangles, but also of the rhombuses

Let’s go by degrees: in what sense is a square also a rectangle? Let’s think well, what is a rectangle? Without going too much in detail we all agree that a rectangle a figure of 4 sides (a quadrilateral) with all the corners of 90 °? But then just ask yourself:

The square Is it a figure of 4 sides with 4 90 ° corners?

The answer is yes, so i squares I am Also rectanglesin practice they are particular rectangles which, unlike the other rectangles, have all 4 equal sides, but they are still rectangles!

Squares and rectangles are relatives also with other geometric figures, in fact they belong to the great family of parallelogramsthat is to say

the parallelograms they are quadrilaterals with i parallel sides Two to two.

Referring to parallelograms we can say, as mathematicians do, that all rectangles, squares inclusive, They are parallelograms with all the straight angles.

But there is more, among the parallelograms we also find i rhombusthat is to say

parallelograms with 4 equal sides.

In the figure above we see 4 rhombusbut two of these also have All the angles of 90 °so they are at the same time both rhombus That rectanglesin other words they are squares!

But then, are the squares rectangles or are they rhombus? They are both the one and the other given both of the characteristic property of the rectangles, have 4 equal cornersthat of that of the rhombuses, to have 4 equal sides: in practice the squares are rhombus-rectangles, or if you prefer rectangles-chasms … and as such they are also parallelograms!

The large family of parallelograms

Things begin to complicate, we make a summary to better understand how mathematicians have classified these geometric figures:

• It starts from the big family of parallelograms: quadrilaterals with i Parallel sides two by two
• The field is narrowed in two different ways, using two different properties:
  1. 1. The parallelograms they have 4 equal corners: i rectangles
  2. 2. The parallelograms they have 4 equal sides: i rhombus
• These two properties are put together and further narrows the field by identifying parallelograms with 4 equal corners And 4 equal sides: i squares.

At each step we restricted the field a little, this means in particular that if it is true that all squares are rectangles, the opposite is not true, in fact there are rectangles that are not square (see figure on the rectangles at the top). In the same way, while all squares are rhombus, there are rhombuses that are not squares (as can be seen in the figure on the rhombuses). In a sense, rectangles And rhombus “Descend” come on parallelograms and i squares “They descend” both from the rhombuses and from the rectangles and therefore also by parallelograms.

But why do mathematicians divide the figures in this strange way? The fact is that in this way a mechanism of heredity of properties which facilitates the study of the figures themselves.

For example, it has been shown that in every rectangle the two diagonals are the same: the squares, which descend from the rectangles, inherit this property and we can say that all squares have the same diagonals without the need to prove it. In practice The squares inherit all the properties of the rectangles And of the rhombus and everything that is shown for rectangles and for the rhombus automatically also applies to squares, without the need for further demonstrations: this leads to a significant work saving for those who study the properties of the different geometric figures. The same is true for any other mathematical concept and this way of classifying is found everywhere in mathematics, and in other sciences!

Equilater triangles are also isoscels

But we remain in the field of geometry and observe the cover image again: how many isoscel triangles are there? The right answer is that all the triangles of the figure are isoscels, even if they are all equilators! For mathematics, in fact, i isoscel triangles I am triangles that have two equal sideswhile i equilateral triangles I am triangles that have three equal sides, So the right question to wonder is:

a equilateral triangle Do you have two equal sides there?

But as we know:

An equilateral triangle even has 3 equal sides, let alone two!

So yes, equilateral triangles are also isoscel triangles.