Earlier in this course, you explored Euclidean geometry, which is the study of flat space. This approach follows the teachings of Euclid, in which he describes the relationships between points, lines, and planes without any numerical measurement. You saw evidence of Euclidean geometry inside several proofs and geometric constructions. In contrast, the focus of this unit is understanding geometry using positions of points in a Cartesian coordinate system. The study of the relationship between algebra and geometry was pioneered by the French mathematician and philosopher René Descartes. In fact, the Cartesian coordinate system is named after him. The study of geometry that uses coordinates in this manner is called analytical geometry.

It’s clear that this course teaches a combination of analytical and Euclidean geometry. Based on your experiences so far, which approach to geometry do you prefer? Why? Which approach is easier to extend beyond two dimensions? What are some situations in which one approach to geometry would prove more beneficial than the other? Describe the situation and why you think analytical or Euclidean geometry is more applicable.

Based on your experiences so far, which approach to geometry do you prefer?

I think that my preference depends on the situation. Euclidean geometry has a basic understanding that makes easy to deal with common life experiences, while analytical geometry request a though more complex and it is used at its best when it comes to resolving mathematical problems in a geometrical way, like functions. 

Why?

Because they have a basic approach that is completely different. Euclidean geometry is based on shapes and numbers, while analytic geometry is based on the relationship between algebra and geometry. One is nearer the human experiences of everyday life, the other is more abstract and so more useful in other contexts. 

Which approach is easier to extend beyond two dimensions?

Intuitively it is easier to extend beyond two dimensions the euclidian approach. All you have to do is to rotate a shape on one of its bases. On the other hand, once you've grasped the concept behind the analytic geometry, perhaps it is easier in that way as you can refer to a geometrical shape in algebra forms.  

What are some situations in which one approach to geometry would prove more beneficial than the other?

everyday life situations, the first form of teaching geometry, basic understanding, visual media are all situations in which Euclidean approach is the best. Complex situation, architecture, informatics, models of prediction, function study, complex mathematics are instead the situations in which analytic approach works the best. 

Describe the situation and why you think analytical or Euclidean geometry is more applicable.

The first form of teaching geometry comes from the recognition of shapes and forms, and it is a concrete subject that could be done with concrete examples. On the other hand, analytic geometry revolves around abstract concepts that should have a solid base before being approached as otherwise, they're not understandable, and so they are used in complex models and mathematics.