To find the length of one of the legs of a 45-45-90 triangle where the hypotenuse is 10 units, we can use the properties of this special type of triangle. In a 45-45-90 triangle, the legs are congruent, and the hypotenuse is √2 times the length of one leg.
Let's denote the length of one leg as [tex]\( x \)[/tex].
Given that the hypotenuse is [tex]\( 10 \)[/tex] units, we can set up the following equation relating the hypotenuse and the leg of the triangle:
[tex]\[ x \sqrt{2} = 10 \][/tex]
To solve for [tex]\( x \)[/tex], we need to isolate [tex]\( x \)[/tex] on one side of the equation. We can do this by dividing both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10}{\sqrt{2}} \][/tex]
Next, we rationalize the denominator. To do this, we multiply both the numerator and the denominator by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = \frac{10 \sqrt{2}}{2} \][/tex]
Simplifying this expression:
[tex]\[ x = 5 \sqrt{2} \][/tex]
Thus, the length of one of the legs of the triangle is [tex]\( 5 \sqrt{2} \)[/tex] units.
To confirm, let's approximate [tex]\( 5 \sqrt{2} \)[/tex]:
[tex]\[ 5 \sqrt{2} \approx 5 \times 1.414 \approx 7.07 \][/tex]
So, the length of one of the legs, when the hypotenuse is 10 units, is approximately [tex]\( 7.07 \)[/tex] units.
From the given options, the correct answer is:
C. [tex]\( 5 \sqrt{2} \)[/tex] units
