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5

Have these students had trigonometry? If so, they may have seen the formulas for rotating a point around the unit circle. $$x' = x\cos(\theta) - y\sin(\theta)$$ $$y' = x\sin(\theta) + y\cos(\theta)$$ If we think of the slope $m=\frac{y}{x}$ relative to an origin of $\langle0,0\rangle$, then $x$ and $y$ can be thought of as just a point that can be ...

5

Here's the algebra-based proof I've used in a college algebra class. Perpendicular lines are defined as meeting at a right angle. Assume that we know the Pythagorean and distance formulas. A possible lemma is that slope of a line indicates how much $y$ increases for a 1-unit increase in $x$ on that line. Given that $m = \Delta y / \Delta x$, when $\Delta x = ... 4 Since you used "$90^\circ$", let me suggest something motivated by trigonometry. Define the relative-slope between two lines by $$m_{rel}=\frac{m_2-m_1}{1+m_2m_1},$$ from the trigonometric identity $$\tan(\theta_2-\theta_1)=\frac{\tan\theta_2-\tan\theta_1}{1+\tan\theta_2\tan\theta_1}.$$ If the lines are parallel [i.e.$\theta_2-\theta_1=0^\circ$],... 3 Let us define two lines$L_1: y=m_1x+b_1$and$L_2: y = m_2x+b_2$to be perpendicular if their intersection exists and forms a right angle. Clearly$m_1 \neq m_2$. Let$P=(x_o,y_o)$be the point of intersection. Then, $$m_1x_o+b_1 = y_o = m_2x_o+b_2$$ Observe$b_2-b_1 = (m_1-m_2)x_o$this will be important later. Furthermore, select$x_2 > x_o\$ and ...

7

These are pretty much universally called lattice paths, which Will Orrick comments on the question. A quick search on the arXiv reveals a large number of papers from combinatorics and computer science using the term in this way. (The main competition is from papers about path integrals on lattices, which gets unfortunately concatenated to "lattice path ...

3

The term that I have always used for this is “rectilinear”, that is, following the lines of a rectangular grid. Most of the definitions of rectilinear found on the internet are commonly non-mathematical and IMHO, not correct (they essentially define it the same as “linear” which is clearly not what it means in math). However the Wikipedia article on ...

10

In the field of micro/nano-lithography, such geometry would be called Manhattan geometry; containing only two directions of edges orthogonal to each other. If diagonals were included, it would be classified as having skew edges. Finally, if it were truly freeform, it would be called curvilinear. Example: https://doi.org/10.1117/12.2243030

6

This paper calls a horizontal/vertical path a rook path, for the movement of a rook on a chess board. If the paths connect the lower-left corner to the upper-right corner, this would be a North-East Lattice Path. Edit: If the path didn't need to go through points with integer coordinates, I would submit the name "Cardinal Paths" in reference to the ...

22

Generally, this math falls under the scope of what is commonly called Taxicab Geometry. I would use taxicab path as a noun to describe the specific paths illustrated in the original question; whereas taxicab geometry would be a term I'd use for the subset of mathematics covering these types of scenarios.

3

AOPS is more comprehensive and mechanical - they don't provide the proofs. The other book is more rigorous as it is proof oriented

4

Have you checked out Math Open Reference? The site has animations that demonstrate how to do the constructions, in the style that you indicated. It won't let students DO the construction online; instead it shows them the steps necessary to do the construction. It also includes printable worksheets for them to practice doing it by hand as well as providing an ...

4

This looks promising but it is not free: mathspad.co.uk.       You can experiment with the tools without creating (or paying for) an account:

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