# In a right triangle, does a+b/c have a name, and how does it change? **URL:** \ **Category:** Factual Questions\ **Created:** [March 30, 2015, 2:44pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336 "2015-03-30T14:44:14Z")\ **Posts on this page:** 7\ **Page:** 1
**Author:** ![Leo\_Bloom](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/leo_bloom/32/10377_2.png) [@Leo\_Bloom](https://boards.straightdope.com/u/Leo_Bloom)\ **Post date:** [March 30, 2015, 2:44pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/1 "2015-03-30T14:44:14Z")
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 a_2 + b_2 = 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. ---
**Author:** ![bup](https://avatars.discourse-cdn.com/v4/letter/b/6bbea6/32.png) [@bup](https://boards.straightdope.com/u/bup)\ **Post date:** [March 30, 2015, 3:05pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/2 "2015-03-30T15:05:48Z")
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. ---
**Author:** ![Lumpy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lumpy/32/446_2.png) [@Lumpy](https://boards.straightdope.com/u/Lumpy)\ **Post date:** [March 30, 2015, 3:30pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/3 "2015-03-30T15:30:37Z")
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) ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [March 30, 2015, 3:43pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/4 "2015-03-30T15:43:29Z")
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_). ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [March 30, 2015, 4:08pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/5 "2015-03-30T16:08:25Z")
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 2c_dc = 2a_dx + 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. ---
**Author:** ![MikeS](https://avatars.discourse-cdn.com/v4/letter/m/919ad9/32.png) [@MikeS](https://boards.straightdope.com/u/MikeS)\ **Post date:** [March 30, 2015, 5:47pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/6 "2015-03-30T17:47:09Z")
> [@Lumpy](#): > > 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) 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. ---
**Author:** ![Ignotus](https://avatars.discourse-cdn.com/v4/letter/i/ecc23a/32.png) [@Ignotus](https://boards.straightdope.com/u/Ignotus)\ **Post date:** [March 30, 2015, 8:08pm UTC](https://boards.straightdope.com/t/in-a-right-triangle-does-a-b-c-have-a-name-and-how-does-it-change/716336/7 "2015-03-30T20:08:55Z")
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.