Question

If c is the measure of the hypotenuse of a right triangle, how do you find each missing measure given a=x-47, b=x, c=x+2?

Answer 1

`a = 2 + 9sqrt(57)`
`b = 49 + 9sqrt(57)`
`c = 51 + 9sqrt(57)`

Explanation:

`a = x - 47`
`b = x`
`c = x + 2`

and we know that the pythagoras theorem is:
`a^2 + b^2 = c^2`

so if we fill that in:
`(x-47)^2 + x^2 = (x+2)^2`

from here we can see that:
`a^2 = x^2 - 94x - 2209`
`b^2 = x^2`
`c^2 = x^2 + 4x + 4`

this leads to:
`2x^2 - 94x - 2209 = x^2 = 4x + 4`
`x62 - 98x - 2216 = 0`

The ABC Formula tells us that in order to calculate `x` we'll use the following method:

`x = (-b ± sqrt(b^2 - 4ac))/(2a)`
with `a = 1`, `b = -98` and `c = -2216`

when we fill that in we get:
`x = (98 + sqrt(-98^2 - (4*-2216)))/(2) = 49 + 9sqrt(57)`
or
`x = (98 - sqrt(-98^2 - (4*-2216)))/(2) = 49 - 9sqrt(57)`

`49 - 9sqrt(57)` equals to a negative number, and since we know that x is the length of side b we know that `x` cannot be negative.
therefore we know that `x = 49 + 9sqrt(57)`

we fill that in for the sides:

`a = x - 47` `->` `a = 2 + 9sqrt(57)`
`b = x -> b = 49 + 9sqrt(57)`
`c = x + 2 -> c = 51 + 9sqrt(57)`

and there's your answer!