To find the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle when the hypotenuse is given, we use the properties of this type of triangle. In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the legs are equal in length and the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times longer than each leg.
Given:
- Hypotenuse = [tex]\(10 \sqrt{5}\)[/tex] inches
Step-by-step solution:
1. Identify the relationship in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle:
[tex]\[
\text{leg} = \frac{\text{hypotenuse}}{\sqrt{2}}
\][/tex]
2. Substitute the given hypotenuse:
[tex]\[
\text{leg} = \frac{10 \sqrt{5}}{\sqrt{2}}
\][/tex]
3. Simplify the expression:
[tex]\[
\frac{10 \sqrt{5}}{\sqrt{2}} = \frac{10 \sqrt{5} \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10 \sqrt{10}}{2} = 5 \sqrt{10}
\][/tex]
4. Thus, the length of one leg of the triangle is:
[tex]\[
5 \sqrt{10}
\][/tex]
Therefore, the correct answer is [tex]\(5 \sqrt{10}\)[/tex].
