Using Geometry and Trigonometry

Question

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.

How does this approach differ from other approaches to geometry that you have used or seen? 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 one approach is more applicable.

Answer 1

The difference between the two approaches comes down to what you are allowed to use as evidence. In the geometry you probably did first, which is usually called synthetic or Euclidean geometry, you work from the picture and from a list of postulates and theorems. If you want to prove two triangles are congruent, you hunt for matching sides and angles and you cite something like SAS. There are no numbers attached to the points at all.

In analytic geometry you drop the figure onto a coordinate grid first, and after that every geometric statement turns into an algebra problem. Two lines being perpendicular stops being a picture fact and becomes the statement that their slopes multiply to −1. The distance between two points stops being something you measure and becomes

d = √((x₂ − x₁)² + (y₂ − y₁)²)

which is really just the Pythagorean theorem wearing coordinates. A circle stops being the set of points a fixed distance from a center and becomes (x − h)² + (y − k)² = r². That is what Descartes gave us, a translation dictionary between shapes and equations.

As for which one I would rather use, it depends on what the problem is asking, and I think that is the honest answer rather than picking a favorite. If a problem gives me specific points or specific lengths, I go analytic every time, because then I am just computing and I cannot be fooled by a picture that is drawn slightly wrong. If the problem is a general claim about all triangles or all parallelograms, synthetic proofs are usually shorter and they show you WHY the result is true instead of just confirming it for one case.

The extension question has a clear answer though. Analytic geometry extends past two dimensions with almost no effort. You add a z and the distance formula becomes √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²), and the same idea keeps working in four dimensions or forty even though nobody can picture it. Synthetic geometry runs into trouble there, because the whole method leans on being able to draw and see the figure, and your intuition about what a picture looks like stops being reliable once you go past three dimensions.

For concrete situations, anything a computer has to do is analytic. Video game graphics, GPS, robotics and mapping software all store points as coordinates and do everything with the formulas above, since a machine cannot look at a diagram and notice that two angles seem equal. On the other side, if you are proving something like the angle bisectors of a triangle always meet at one point, doing it synthetically is far cleaner than assigning coordinates to everything and grinding through the algebra.

So the short version is that synthetic geometry is better for understanding and proving general truths, analytic geometry is better for calculating specific answers and for anything that has to scale up in dimensions or run on a computer. Most of the time you end up using both.