To determine the length of one of the legs of an isosceles right triangle when the hypotenuse is 5 centimeters, follow these steps:
1. Understand the properties of an isosceles right triangle:
- In an isosceles right triangle, the two legs are of equal length.
- The relationship between the legs and the hypotenuse can be derived from the Pythagorean theorem:
[tex]\[
\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2
\][/tex]
This simplifies to:
[tex]\[
2 \cdot \text{leg}^2 = \text{hypotenuse}^2
\][/tex]
2. Substitute the given hypotenuse:
- The hypotenuse is given as 5 centimeters:
[tex]\[
2 \cdot \text{leg}^2 = 5^2
\][/tex]
[tex]\[
2 \cdot \text{leg}^2 = 25
\][/tex]
3. Solve for the leg:
- Divide both sides of the equation by 2 to solve for [tex]\(\text{leg}^2\)[/tex]:
[tex]\[
\text{leg}^2 = \frac{25}{2}
\][/tex]
[tex]\[
\text{leg} = \sqrt{\frac{25}{2}}
\][/tex]
4. Simplify the expression for the leg:
- Simplify the square root:
[tex]\[
\text{leg} = \frac{5}{\sqrt{2}}
\][/tex]
5. Rationalize the denominator:
- Multiply both the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
\text{leg} = \frac{5 \sqrt{2}}{2}
\][/tex]
Therefore, the length of one of the legs of the isosceles right triangle is [tex]\(\frac{5 \sqrt{2}}{2}\)[/tex] centimeters, which corresponds to the correct answer.
