Question

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

Answer 1

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=(-b+/-√(b^2-4ac))/(2a)`
so for
`x^2-12x-288=0`, a=1, b=-12, c=-288
subsituting

`x=(-(-12)+/-√((-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

Answer 2

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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