To determine the length of the hypotenuse of triangle DEF, where each leg has a length of 36 units, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
For our triangle DEF, since it is a right triangle with both legs equal (36 units each), we apply the theorem as follows:
[tex]\[ a = 36 \text{ units} \][/tex]
[tex]\[ b = 36 \text{ units} \][/tex]
The theorem is written as:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
Substituting the known values:
[tex]\[ c^2 = 36^2 + 36^2 \][/tex]
This simplifies to:
[tex]\[ c^2 = 1296 + 1296 \][/tex]
[tex]\[ c^2 = 2592 \][/tex]
To find [tex]\( c \)[/tex], we take the square root of both sides:
[tex]\[ c = \sqrt{2592} \][/tex]
Simplifying the square root of 2592, we get:
[tex]\[ c = 36\sqrt{2} \][/tex]
Therefore, the length of the hypotenuse of triangle DEF is:
[tex]\[ 36\sqrt{2} \text{ units} \][/tex]
Given the multiple-choice options:
- 18 units
- [tex]\( 18\sqrt{2} \)[/tex] units
- 36 units
- [tex]\( 36\sqrt{2} \)[/tex] units
The correct answer is [tex]\( 36\sqrt{2} \)[/tex] units.
