To find the length of the hypotenuse in a 45-45-90 triangle where one leg is given as 6, we follow these steps:
1. Identify the properties of a 45-45-90 triangle:
- A 45-45-90 triangle is an isosceles right triangle, meaning the two legs are of equal length.
- The relationship between the legs [tex]\( a \)[/tex] and the hypotenuse [tex]\( c \)[/tex] in such a triangle is given by [tex]\( c = a\sqrt{2} \)[/tex].
2. Given information:
- The length of one leg is 6.
3. Applying the properties:
- Let the length of the leg be [tex]\( a = 6 \)[/tex].
4. Calculate the hypotenuse:
- Use the formula for the hypotenuse [tex]\( c \)[/tex] in a 45-45-90 triangle: [tex]\( c = a\sqrt{2} \)[/tex].
- Substitute [tex]\( a = 6 \)[/tex] into the formula: [tex]\( c = 6\sqrt{2} \)[/tex].
5. Simplify the expression:
- The hypotenuse length in exact form is [tex]\( 6\sqrt{2} \)[/tex].
6. Numerical value:
- Using [tex]\( \sqrt{2} \approx 1.414213562 \)[/tex], we get the numerical value for the hypotenuse, which is:
[tex]\( 6 \times 1.414213562 = 8.485281374 \)[/tex].
Therefore, the length of the hypotenuse, expressed in simplest radical form, is [tex]\( 6\sqrt{2} \)[/tex].
Thus, the correct answer from the provided options is:
[tex]\[ 6\sqrt{2} \][/tex]
