100 POINTS 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.

THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. A plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? Explain your reasoning.

That's to illustrate what this is talking about. So in spiritual geometry, Uh, instead of working with the plane, we're working on a spear, so all of the ideas of a plane would be the same thing as the spear. Uh, anything with a line. Uh, we would call it a great circle, so every line is represented by the same circle. So if we were to draw a triangle, we would need three lines. Uh, it's Ah, no. How could I do it? So let's see here. You can imagine circle on like that. So then if you think about the front side so that that would be one front side of a circle, that would be another front side, and then he would be a third. So where the three sides would cross would give us like the triangle. Imagine that being on the front side of that sphere. Um, now, for a similar triangle, the angles always have to be the same. So this idea of an angle at each one of these positions ah, would have to remain the same as we make a triangle bigger. But to make a triangle bigger, we can't change the size of these circles. So a line is defined according to this problem as being what I would call a great circle like it's always the same size as the radius of the sphere. And so if we move one of those, even just one of them, it would change the angles of the others. So in order to make this triangle bigger, uh, like quote unquote bigger are angles would have to necessarily change as well. So if the angles air the same one interesting thing that comes out of this is that if the angles of the same then the triangles are congruent so in spherical geometry there are no similar triangles. So the answer to this would be No. There are no similar triangles on bats because the T B similar has to be different size with the same angles. And we cannot keep the angles the same if we change the size. Okay, I'll play. That explained it pretty well. And I'm sorry about my picture. It's three D is always kind of hard to draw. Um, and and again, all of this comes from the definition of the line for a spherical geometry. Being one of these great circles that has to have the same radius as the original sphere. Okay. Thanks again

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