Question

The sides of a right triangle are x, x+21, and x+27. what is the length of the hypotenuse

Answer 1

Answer: 51 uinits

The three numbers are x , x + 21 and x + 27. 
The greatest of the 3 numbers is x + 27, so x + 27 has to be the hypotenuse.

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Pythagoras Theorem
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a² + b²  = c²

x² + (x+21)² = (x+27)²

x² + x² + 42x + 441 = x² + 54x + 729

x² - 12x -288 = 0

(x - 24)(x + 12) = 0

x = 24 or x = -12 (rejected, length cannot be negative)

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Find hypotenuse
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 Hypotenuse = x + 27 = 24 + 27 = 51 units

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Answer: 51 units
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Answer 2

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 $a^2+b^2=c^2$

we can see that x+27 is bigger than the other 2 numbers so x+27 must be the length of the hyptonuse

therefor
$(x)^2+(x+21)^2=(x+27)^2$
expand
$x^2+x^2+42x+441=x^2+54x+729$
add like terms
$2x^2+42x+441=x^2+54x+729$
subtract $x^2+54x+729$ from both sides
$x^2-12x-288=0$
solve, use quadratic formula
for $ax^2+bx+c=0$, $x=\frac{-b+/-\sqrt{b^2-4ac}}{2a}$
so for $x^2-12x-288=0$, a=1, b=-12, c=-288
subsituting
$x=\frac{-(-12)+/-\sqrt{(-12)^2-4(1)(-288)}}{2(1)}$
$x=6+/-18$
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