In a right triangle, does a+b/c have a name, and how does it change?

Factual Questions
1

I think about this constantly: rather than go west and north from the dog park, I cut across on a straight line to get to the same point.

If I walk 3 blocks west and 4 blocks north, I “save” 2/5 of a block with the shortcut.

  1. Does that “savings” have a name?
  2. Given the Pythagorean theorem a2 + b2 = c*2, how does c change (its regular increase) if a and b each are increased by 1 in succession?

I feel like question 2) would be on a quiz that everyone else would get but me.

3 (less important): Is “monotonically” a word that fits here?

I blame my junior high school math teacher.

2

Question 1 - I don’t know.

I think the formula you want, though, is c[sup]2[/sup] / (a[sup]2[/sup] + b[sup]2[/sup])

The ‘savings’ is maximized in a 45/45/90 triangle, when it’s 2[sup].5[/sup] / 2.

The savings by increasing sides a and b by 1 each is entirely based on the angles A and B.

3

Isn’t this trigonometry? If the hypotenuse is C, and the adjacent and opposite sides A and B, then the cosine is A/C and the sine B/C; so the ratio of (A+B)/C should be the cosine plus the sine (cos+sin)

4

This looks like a special case of the* triangle inequality*.

In vector notation, ||a|| + ||b|| >= ||**a **+ b||. (Equality holds when a or b are zero or they are parallell, the latter ruled out in your case).

I don’t know of any special name for the function f(x,y) = x + y - sqrt( x^2 + y^2), but for non- negative x and y this function is certainly monotonic (non-decreasing) with respect to both variables. (For positive values of both x and y, it’s actually strictly increasing).

5

For 2) you can use a little calculus. As c[sup]2[/sup] = a[sup]2[/sup]+b[sup]2[/sup], for a small increase x in both a and b, we’d have

2cdc = 2adx + 2b*dx

=> dc/dx = (a+b)/c

We must have a+b > c so c increases more than the common increase in both a and b. This ratio is minimal when a or be is zero and c increases by the same amount as do a and b. The ratio is maximal when a = b = c/sqrt(2). In that case c increases by approximately sqrt(2) for each unit increase in a and b.

6

This is what I came in here to say. Note also that if you apply various trigonometric identities, this quantity is equal to

cos(theta) + sin(theta) = √2 cos(theta - 45°).

This is obviously maximized (i.e., for a given “diagonal distance”, the ratio of “rectangular distance” to “diagonal distance” is largest) when theta = 45°, in which case the ratio is √2.

As far as “monotonically” goes: it depends on what you’re referring to. It is true that increasing either leg of the right triangle necessarily causes an increase in the length of the hypotenuse. So in this sense, the hypotenuse is a monotonic function of the lengths of the legs.

7

Sorry if I have caused confusion; I didn’t really get your question (or rather, I jumped to conclusions from the outset of it)

What you’re measuring is not the difference in distance; it’s the relative difference in distance. So, instead of the function x+y-sqrt(x^2+y^2), we should consider the function (x+y-sqrt(x^2+y^2))/sqrt(x^2+y^2).

And this function is not monotonic; for positive x,y ´s it has a maximum at x=y.