proportional relationships are those that are the same in size, shape, and proportion. For instance, a right triangle is proportional to a square.
In a right triangle the top side is the same as the bottom side. For instance, a right triangle is a square. A square is a proportional relationship because we can think of all of its sides as being the same size.
The biggest constant in math is proportion. We can think of proportions only in terms of equal length. So a square won’t be the same shape as a rectangle. We can’t imagine a square as being the same size as a rectangle. We can only imagine a rectangle as being the same size as a square.
That’s why we have angles. For example, a triangle is proportional because the sides are the same size. The sides of a square are equal length. A square is a proportional relationship because the bigger side is the same length as the smaller side.
It is a fact that a square is a proportional relationship because the side of the square is the same size as the side of a rectangle. But what is a rectangle and why does it matter that we have to look at it that way? The answer is the definition of proportional relationship. A rectangle is defined as the area of one side times the length of the other. So, for a rectangle to be a proportional relationship, the length of the longer side must be the same as the shorter side.
This is a little confusing. If you know what a proportional relationship looks like, you would think that if the length of the longer side is the same as the shorter side, the rectangle is a proportional relationship. But that would be wrong because the lengths of the longer side and the shorter side are not the same size.
As a matter of fact, when you multiply side length times area, you get the formula for proportion. This, however, is not the case with rectangles. Rectangles do not have the same length. So, if you want a rectangle to be proportional, you would have to say the length of the shorter side is the same as the longer side. This may sound right, but it isn’t true.
The thing that I want to discuss is the definition of proportional relationship. The first definition I encountered was in Euclid’s Elements.
Euclides Elements is one of the books that I read as a child. It was very influential, and is what gave me an understanding of proportions and geometry. Basically, Euclides Elements is a book of very simple geometric diagrams that I was able to find and understand. The book, however, also contains the definition of proportion. The first definition was given by one of the main characters. The first definition I found online was this one by a guy named Huygens.
The definition of proportion and the concept of “proportional relationship” is an important concept in math, but its significance goes far beyond that. It’s a concept that’s used extensively in various disciplines. We could talk about the importance of proportion, but the concept itself should be important to every student of math.