This is a bit of a can of worms. Let's unpack a little.
We were given no additional information other than that stated in the page.
This is your daughter's homework, not yours. Be careful with this distinction, because coloring a child's question with adult interpretations can lead to trouble (and does in this case).
it is not stated whether we are ...
The point of compass-and-straightedge constructions is for the students to get experience reasoning about axiomatic systems. The accuracy of the actual drawings is basically irrelevant (as long as it's not so sloppy that it impedes visualization) — the point is that they can prove the accuracy of the idealized constructions.
This method of axiomatic ...
(Disclaimer: I peronally really don't care if one uses $\tau$ or $\pi$, both are just numbers for me.)
But I would strongly recommend to use $\pi$. Why?
In every technical literature, in many popular literature, the people always use $\pi$ (even worse: $\tau$ is used for different things than $\tau=2\pi$ which would confuse when reading those literature). ...
Native Peruvian architecture makes heavy use of the trapezoid for stability in earthquakes. (The Spaniards thought they were primitive as they didn't use arches ... but most of the Spanish buildings have collapsed or had to be rebuilt).
It's especially apparent in their doorways and windows.
Other examples with licensing such that I ...
I feel that it is perhaps a little irresponsible to teach $\tau$ instead of $\pi$. As a first introduction, it is the norm which should be taught: teaching a rare alternative to $\pi$ only serves to confuse students, especially when almost all available resources use $\pi$ instead of $\tau$. Imagine a student's confusion when they see $\tau$ in class and $\...
Given that I had a good time coercing Mathematica to give me the pictures below I might as well promote my comment to an answer.
My favorite such a result is the so called Steiner's porism.
It is a statement about the relation of two circles, one inside the other. Such as these two:
We get a so called Steiner chain of circles from this by first drawing a ...
I cannot answer the OP's question about cross-cultural/international perspectives, but here is a historical perspective that may be helpful. The issue here (whether the category "rectangles" includes or excludes the category "squares") is one aspect of a larger question having to do with whether the classification of quadrilaterals should be partitional or ...
One should teach $\pi$. One might discuss that there is a choice that is made that is somewhat arbitrary, and there are also reasons for a different choice but I do not see this as that relevant to make much ado about this.
It should perhaps also be noticed that there are two conflicting proposals for $\tau$. Eagle (1958) proposed $\pi/2$ and Palais (2001) ...
Here is a simple construction. Adjust to taste.
1. Draw a line $l$ passing through a point O.
2. Construct circles of radius 3 and 4 with centre O. Call them $C_3$ and $C_4$. Let the intersection of $C_4$ and $l$ be A. (Note that OA is 4 units.)
3. Construct an interval of length 5 from A to $C_3$. Call it B. (Note that OB is 3 units, and AB is 5 units.)
The Chinese came up with the following a long time ago. Probably something better, but this is the gist of it.
Let's start with a right triangle with height b=4 and base a=3. We know it has some hypotenuse, c, but we don't know it's length because that's what we're trying to prove.
Now we're going to make 3 copies, and rotate them each 90°.
Next we're ...
I'd like to tackle the question from another point of view than JPBurkes answer: If you accept, that mathematical argumentation (whatever level) is an essential part of mathematics courses in K-12, than Euclidean Geometry is a great way to implement this:
Visuality Euclidean Geometry deals with objects that can be easily visualized. It can be properly ...
TL;DR: It's not the triangles that are interesting; it's the
mathematical concepts that can best be explained by using one of the
most primitive geometrical shapes.
The reason for intensive use of triangles goes beyond knowledge about triangles per se.
It's the act of and steps in proving a theorem that's important to learn at this stage - start with ...
A corkscrew (for a helix):
A donut (for a torus):
A football (for a spheroid)
And then, there's also the atomium (for which I am not sure exists a geometric name)
cooling towers (for a hyperboloid)
and the pentagon (well, for a pentagon):
A pyramid is, of course, a pyramid.
Lastly, a soccer ball is a truncated icosahedron
(Images by wiki, pedia)
The hexagon at the north pole of Saturn:
It is known that
"[regular shapes] form in an area of turbulent flow between ... two different rotating fluid bodies with dissimilar speeds."
and this has been proposed as an explanation for the phenomenon.
Incidentally, the Earth could easily fit inside the pole hexagon.
An article in ...
The fact that there is a 3-4-5 triangle that is a right triangle is unique to the Euclidean plane. There is no such triangle in the spherical or hyperbolic planes. Since the Pythagorean theorem is equivalent to the parallel postulate, any proof that a 3-4-5 triangle is a right triangle will somehow depend on the Pythagorean theorem/parallel postulate.
I'd say different proofs usually employ different techniques, which in turn might be applicable to different sets of other theorems. So the more proofs I know for one theorem, the higher the chances that I'll be able to adapt at least one of them to a similar (or maybe not so similar) theorem I'm trying to prove.
Furthermore, seeing several techniques ...
One angle you could look at is molecular geometry. Not really my subject area but a couple of examples:
Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as ...
$\tau$ should be taught in schools
There's plenty of material arguing why $\tau$ is a much more intuitive and easier to teach concept (some of my favorites: 1,2,3) and I don't want to rehash their arguments, but if you think that $\tau$ is just to make equations look nicer, please check out those resources.
The question at hand is whether math educators ...
"Turning Torso," an apartment building in Malmö, Sweden designed by
architect Santiago Calatrava, following a twisting spiral.
It consists of
"nine segments of five-story pentagons that twist relative to each other as it rises; the topmost segment is twisted 90 degrees clockwise with respect to the ground floor."
Graph paper (or square floor tiling) to the rescue!
Proof by picture for a 3 4 5 triangle:
Because the drawing is on the grid and not the skew tiling of the square on the hypotenuse, determining the area is not inscrutable. The blue square (it is a square by adding angles of the triangle) is area 25 by adding the four blue triangles (obviously 6 each) and ...
Of course there is the deep (~2000 yrs) history of compass/straightedge constructions.
Another more modern alternative is origami constructions:
Robert Lang, "Origami and Geometric Constructions," 1996.
In particular, one can trisect an angle via origami folding
(under, e.g., Huzita's origami axioms):
Figure from Geometric Folding ...
I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math:
Skipping over the motivation doesn't make something easier to understand.
As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the ...
One other poster mentioned arches; I'd like to add in the Gothic arch as an example of circular segments. These are great examples of arcs as well. I find them much more interesting, and they don't always have to have the angle shown here; the location of the circle's center can vary depending on the "slope" of the arch that is desired. There are also three- ...