To determine the length of the hypotenuse of triangle DEF, which is a right-angle triangle with both legs having equal lengths, we will use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angle triangle:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the legs, and [tex]\( c \)[/tex] is the length of the hypotenuse.
In this problem, both legs [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are given as 36 units each. Therefore, we can write:
[tex]\[ a = 36 \][/tex]
[tex]\[ b = 36 \][/tex]
Substituting these values into the Pythagorean theorem:
[tex]\[ 36^2 + 36^2 = c^2 \][/tex]
First, calculate [tex]\( 36^2 \)[/tex]:
[tex]\[ 36^2 = 1296 \][/tex]
Substitute [tex]\( 1296 \)[/tex] for both [tex]\( 36^2 \)[/tex] values:
[tex]\[ 1296 + 1296 = c^2 \][/tex]
Add the two values:
[tex]\[ 2592 = c^2 \][/tex]
To find [tex]\( c \)[/tex], we take the square root of both sides:
[tex]\[ c = \sqrt{2592} \][/tex]
We know that:
[tex]\[ \sqrt{2592} \approx 50.91168824543143 \][/tex]
Therefore, the length of the hypotenuse is approximately [tex]\( 50.91168824543143 \)[/tex] units.
From the provided options, the answer that best matches this value in simpler form is:
[tex]\[ 36 \sqrt{2} \][/tex]
Hence, the correct answer is:
[tex]\[ 36 \sqrt{2} \][/tex] units
