To find the difference between the two possible lengths of the third side of a right triangle when given two side lengths, we need to consider two scenarios: when one given side is the hypotenuse and when the hypotenuse is the third side.
### Step-by-Step Solution:
1. Identify the given sides: We have two sides of the triangle, 5 inches and 8 inches.
2. Calculate the hypotenuse when the given sides are the legs:
To find the hypotenuse [tex]\( c \)[/tex] when 5 inches and 8 inches are the legs [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we use the Pythagorean theorem:
[tex]\[
c = \sqrt{a^2 + b^2}
\][/tex]
[tex]\[
c = \sqrt{5^2 + 8^2}
\][/tex]
[tex]\[
c = \sqrt{25 + 64}
\][/tex]
[tex]\[
c = \sqrt{89}
\][/tex]
The calculated length of the hypotenuse is approximately 9.4 inches.
3. Calculate the length of the other side when one of the given sides is the hypotenuse:
In this case, we assume one of the given sides (say 8 inches) is the hypotenuse and we find the length of the other leg [tex]\( a \)[/tex] when the known leg [tex]\( b \)[/tex] is 5 inches.
Using the Pythagorean theorem in reverse:
[tex]\[
a = \sqrt{c^2 - b^2}
\][/tex]
[tex]\[
a = \sqrt{8^2 - 5^2}
\][/tex]
[tex]\[
a = \sqrt{64 - 25}
\][/tex]
[tex]\[
a = \sqrt{39}
\][/tex]
The calculated length of the other leg is approximately 6.2 inches.
4. Calculate the difference between the two possible lengths:
There are two possible lengths for the third side: approximately 9.4 inches (when 5 and 8 are legs) and approximately 6.2 inches (when 8 is the hypotenuse).
The difference between these two lengths is:
[tex]\[
\text{Difference} = |9.4 - 6.2|
\][/tex]
[tex]\[
\text{Difference} = 3.2
\][/tex]
5. Round the difference to the nearest tenth:
The difference is already rounded to the nearest tenth.
Thus, the difference between the two possible lengths of the third side of the triangle is 3.2 inches.
