Answer:
Yes!
Step-by-step explanation:
We know that right triangles follow the Pythagorean Theorem, where

So we will put our side lengths into this formula, keeping in mind that c = the hypotenuse, which is the longest side. a and b are fairly arbitrary.

If this works, we know it's a right triangle.
1089 + 1936 = 3025
3025 = 3025
It worked! It's a right triangle!!
The answer is a.) 165 sorry if that’s wrong but i hope it’s right
let's firstly convert the mixed fraction to improper fraction and then take it from there, keeping in mind that the whole is "x".
![\stackrel{mixed}{5\frac{5}{6}}\implies \cfrac{5\cdot 6+5}{6}\implies \stackrel{improper}{\cfrac{35}{6}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{7}{3}x~~ = ~~5\frac{5}{6}\implies \cfrac{7}{3}x~~ = ~~\cfrac{35}{6}\implies 42x=105\implies x=\cfrac{105}{42} \\\\\\ x=\cfrac{21\cdot 5}{21\cdot 2}\implies x=\cfrac{21}{21}\cdot \cfrac{5}{2}\implies x=1\cdot \cfrac{5}{2}\implies x=2\frac{1}{2}](https://tex.z-dn.net/?f=%5Cstackrel%7Bmixed%7D%7B5%5Cfrac%7B5%7D%7B6%7D%7D%5Cimplies%20%5Ccfrac%7B5%5Ccdot%206%2B5%7D%7B6%7D%5Cimplies%20%5Cstackrel%7Bimproper%7D%7B%5Ccfrac%7B35%7D%7B6%7D%7D%20%5C%5C%5C%5C%5B-0.35em%5D%20~%5Cdotfill%5C%5C%5C%5C%20%5Ccfrac%7B7%7D%7B3%7Dx~~%20%3D%20~~5%5Cfrac%7B5%7D%7B6%7D%5Cimplies%20%5Ccfrac%7B7%7D%7B3%7Dx~~%20%3D%20~~%5Ccfrac%7B35%7D%7B6%7D%5Cimplies%2042x%3D105%5Cimplies%20x%3D%5Ccfrac%7B105%7D%7B42%7D%20%5C%5C%5C%5C%5C%5C%20x%3D%5Ccfrac%7B21%5Ccdot%205%7D%7B21%5Ccdot%202%7D%5Cimplies%20x%3D%5Ccfrac%7B21%7D%7B21%7D%5Ccdot%20%5Ccfrac%7B5%7D%7B2%7D%5Cimplies%20x%3D1%5Ccdot%20%5Ccfrac%7B5%7D%7B2%7D%5Cimplies%20x%3D2%5Cfrac%7B1%7D%7B2%7D)
F(x)=(-2/((x+y-2)^(1/2))-(x+y+2)^(1/2)
the only irrational part of this expression is the (x+y-2)^(1/2) in the denominator, so, to rationalize this, you multiply the numerator and denominator by the denominator, as well as the other parts of the expression
also, you must multiply the -sqrt(x+y+2) by sqrt(x+y-2)/sqrt(x+y-2) to form a common denominator
(-2)/(x+y-2)^(1/2)-(x+y+2)^(1/2)(x+y-2)^(1/2)/(x+y-2)^(1/2)
(common denominator)
(-2-(x^2+xy+2x+xy+y^2+2y-2x-2y-4))/(x+y-2)^(1/2)
(FOIL)
(-2-x^2-y^2-2xy+4)/(x+y-2)^(1/2)
(Distribute negative)
(-x^2-y^2-2xy+2)/(x+y-2)^(1/2)
(Simplify numerator)
(-x^2-y^2-2xy+2)(x+y-2)^(1/2)/(x+y-2)^(1/2)(x+y-2)^(1/2)
(Rationalize denominator by multiplying both top and bottom by sqrt)
(-x^2-y^2-2xy+2)((x+y-2)^(1/2))/(x+y-2)
(The function is now rational)
=(-x^2-y^2-2xy+2)(sqrt(x+y-2))/(x+y-2)