To solve this problem, we'll use the Pythagorean Theorem, which states that in a right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. The Pythagorean Theorem is expressed as:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
where [tex]\( c \)[/tex] is the length of the hypotenuse, and [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lengths of the two legs.
Given:
[tex]\[ c = 12 \, \text{cm} \][/tex]
[tex]\[ a = 6 \, \text{cm} \][/tex]
We need to find the length of the unknown leg [tex]\( b \)[/tex].
Step-by-step solution:
1. Start with the Pythagorean Theorem:
[tex]\[ a^2 + b^2 = c^2 \][/tex]
2. Substitute the known values into the equation:
[tex]\[ 6^2 + b^2 = 12^2 \][/tex]
[tex]\[ 36 + b^2 = 144 \][/tex]
3. Isolate [tex]\( b^2 \)[/tex] by subtracting 36 from both sides:
[tex]\[ b^2 = 144 - 36 \][/tex]
[tex]\[ b^2 = 108 \][/tex]
4. Take the square root of both sides to solve for [tex]\( b \)[/tex]:
[tex]\[ b = \sqrt{108} \][/tex]
[tex]\[ b \approx 10.39 \, \text{cm} \][/tex]
So, the length of the unknown leg is approximately [tex]\( 10.39 \, \text{cm} \)[/tex].
Therefore, the correct answer is:
[tex]\[ 10.39 \, \text{cm} \][/tex]
