To find the length of the hypotenuse of a right triangle when the lengths of the legs are known, we use the Pythagorean theorem. The Pythagorean theorem states:
[tex]\[ c = \sqrt{a^2 + b^2} \][/tex]
where [tex]\( c \)[/tex] is the length of the hypotenuse, and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the legs.
Given the lengths of the legs:
[tex]\[ a = 4 \][/tex]
[tex]\[ b = 8 \][/tex]
Substitute these values into the formula:
[tex]\[ c = \sqrt{4^2 + 8^2} \][/tex]
Calculate the squares of the legs:
[tex]\[ 4^2 = 16 \][/tex]
[tex]\[ 8^2 = 64 \][/tex]
Add the squares:
[tex]\[ 16 + 64 = 80 \][/tex]
Take the square root of the sum:
[tex]\[ c = \sqrt{80} \][/tex]
Simplify the square root (if possible):
[tex]\[ \sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5} = 4 \sqrt{5} \][/tex]
However, in this particular question, we already know that the accurate numerical value of [tex]\(\sqrt{80}\)[/tex] is approximately:
[tex]\[ \sqrt{80} \approx 8.94427190999916 \][/tex]
Hence, given approximate numerical result confirms and the closest match to the given options.
Among the provided choices, none of the options explicitly states 8.94427190999916. However, the derived value of the hypotenuse from the calculation [tex]\(\sqrt{80}\)[/tex] correctly simplifies to [tex]\(4 \sqrt{5}\)[/tex].
Therefore, the correct answer is:
A. [tex]\(4 \sqrt{5}\)[/tex]
