To find the length of one leg of an isosceles right triangle where the hypotenuse is [tex]\( 5 \)[/tex] centimeters, we can use the properties of this type of triangle.
An isosceles right triangle has two legs of equal length, and the hypotenuse is opposite the right angle. According to the Pythagorean theorem:
[tex]\[ a^2 + a^2 = c^2 \][/tex]
where [tex]\( a \)[/tex] is the length of both legs and [tex]\( c \)[/tex] is the length of the hypotenuse.
Given that [tex]\( c = 5 \)[/tex]:
[tex]\[ a^2 + a^2 = 5^2 \][/tex]
This simplifies to:
[tex]\[ 2a^2 = 25 \][/tex]
Next, solve for [tex]\( a^2 \)[/tex]:
[tex]\[ a^2 = \frac{25}{2} \][/tex]
Now take the square root of both sides to find [tex]\( a \)[/tex]:
[tex]\[ a = \sqrt{\frac{25}{2}} \][/tex]
Simplify the expression under the square root:
[tex]\[ a = \sqrt{\frac{25}{2}} = \sqrt{12.5} \][/tex]
The value of [tex]\( \sqrt{12.5} \)[/tex], given in the numerical form, is approximately:
[tex]\[ a \approx 3.5355339059327378 \][/tex]
This value matches the expression:
[tex]\[ a = \frac{5 \sqrt{2}}{2} \][/tex]
Hence, the length of one of the legs of the isosceles right triangle is:
[tex]\[ \boxed{\frac{5 \sqrt{2}}{2}} \][/tex]
