To determine the length of one leg in a 45-45-90 triangle given that the hypotenuse is 10 units, we need to use the properties of this specific type of triangle. In a 45-45-90 triangle, the sides are in a specific ratio:
- Both legs are of equal length.
- The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Let's denote the length of each leg as [tex]\( x \)[/tex]. Thus, according to the triangle's properties:
[tex]\[ \text{Hypotenuse} = x \sqrt{2} \][/tex]
Given that the hypotenuse is 10 units, we can set up the equation:
[tex]\[ 10 = x \sqrt{2} \][/tex]
To solve for [tex]\( x \)[/tex], we divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10}{\sqrt{2}} \][/tex]
To rationalize the denominator, we multiply the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} \][/tex]
Since [tex]\( \sqrt{2} \cdot \sqrt{2} = 2 \)[/tex], the equation simplifies to:
[tex]\[ x = \frac{10 \sqrt{2}}{2} \][/tex]
[tex]\[ x = 5 \sqrt{2} \][/tex]
So, the length of one of the legs of the triangle is:
[tex]\[ 5 \sqrt{2} \text{ units} \][/tex]
This corresponds to option C. Therefore, the correct answer is:
C. [tex]\( 5 \sqrt{2} \)[/tex] units
