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?

1 Answer
Feb 8, 2017

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!