the lengths of the two sides of a right triangle are 5 inches and 8 inches . what is the difference between the two possible lengths of the third side of the triangle? round your answer to the nearest tenth

Answer:
The difference between the two possible lengths is 3.2 in

Explanation:
Assume that the sides of the triangle are:
a = 5
b = 8
c = x

First, we will assume that the third side is the hypotenuse:
For the triangle to be right angled:
c^2 = a^2 + b^2
Substitute with the values of a, b and c in the equation and solve for x as follows:
c^2 = a^2 + b^2
x^2 = (5)^2 + (8)^2
x^2 = 89
either x = 9.433 in
or x = -9.433 in (refused as no length is in negative)
So, first possible value of the third side is 9.433 in

Second, we will assume that the 8 in is the hypotenuse of the triangle.
For the triangle to be right angled:
b^2 = a^2 + c^2
(8)^2 = (5)^2 + x^2
64 = 25 + x^2
x^2 = 64 - 25 = 39
either x = 6.245 in
or x = -6.245 in (refused as no length is in negative)
Therefore, the second possible value of the third side is 6.245 in

Finally, we will get the difference between the two length as follows:
difference = 9.433 - 6.245 = 3.188 which is approximately 3.2 in

Note:
We cannot assume that the 5 in is the hypotenuse because in the right-angled triangle the hypotenuse is the longest side. We are given that one side = 8 in therefore, it is impossible for the 5 in to be the hypotenuse.

Hope this helps :)