Certainly! Let's solve this problem step-by-step.
We are given a 45°-45°-90° triangle, also known as an isosceles right triangle. In such a triangle, the lengths of the legs are equal, and the hypotenuse is related to the legs by a specific ratio.
Given:
- The hypotenuse measures [tex]\( 10 \sqrt{5} \)[/tex] inches.
We need to find the length of one leg.
In a 45°-45°-90° triangle, the relationship between the lengths of the legs and the hypotenuse is given by:
[tex]\[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} \][/tex]
Thus, to find the length of one leg ([tex]\( L \)[/tex]), we rearrange the formula:
[tex]\[ L = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Plugging in the given hypotenuse:
[tex]\[ L = \frac{10 \sqrt{5}}{\sqrt{2}} \][/tex]
To simplify the expression, we rationalize the denominator:
[tex]\[ L = \frac{10 \sqrt{5}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \][/tex]
[tex]\[ L = \frac{10 \sqrt{5} \cdot \sqrt{2}}{2} \][/tex]
[tex]\[ L = \frac{10 \sqrt{10}}{2} \][/tex]
[tex]\[ L = 5 \sqrt{10} \][/tex]
Therefore, the length of one leg of the triangle is [tex]\( 5 \sqrt{10} \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{5 \sqrt{10}} \][/tex]
