hypotonuse is the longest side
pythagorean theorem tells us that if the lengths of the legs of a right triangle are a and b and the hyptonuse is c then [tex]a^2+b^2=c^2[/tex]
we can see that x+27 is bigger than the other 2 numbers so x+27 must be the length of the hyptonuse
therefor
[tex](x)^2+(x+21)^2=(x+27)^2[/tex]
expand
[tex]x^2+x^2+42x+441=x^2+54x+729[/tex]
add like terms
[tex]2x^2+42x+441=x^2+54x+729[/tex]
subtract [tex]x^2+54x+729[/tex] from both sides
[tex]x^2-12x-288=0[/tex]
solve, use quadratic formula
for [tex] ax^2+bx+c=0[/tex], [tex]x=\frac{-b+/-\sqrt{b^2-4ac}}{2a}[/tex]
so for [tex]x^2-12x-288=0[/tex], a=1, b=-12, c=-288
subsituting
[tex]x=\frac{-(-12)+/-\sqrt{(-12)^2-4(1)(-288)}}{2(1)}[/tex]
[tex]x=6+/-18[/tex]
x=-12 or 24
x can't be negative since we can't have a negative side ength
x=24
x+27 is hypotonuse
x+27=24+27=51
the hypotonuse is 51 units long
