To solve this problem, we need to understand the properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle. In such a triangle, the legs are of equal length, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Given:
- The hypotenuse ([tex]\(c\)[/tex]) measures [tex]\(7 \sqrt{2}\)[/tex] units.
To find the length of one leg ([tex]\(a\)[/tex]), we use the relationship between the hypotenuse and the legs in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, which is:
[tex]\[ c = a \sqrt{2} \][/tex]
Substituting the given hypotenuse length into the equation:
[tex]\[ 7 \sqrt{2} = a \sqrt{2} \][/tex]
To isolate [tex]\(a\)[/tex], we divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ a = \frac{7 \sqrt{2}}{\sqrt{2}} \][/tex]
The [tex]\( \sqrt{2} \)[/tex] terms cancel out, leaving:
[tex]\[ a = 7 \][/tex]
Thus, the length of one leg of the triangle is 7 units.
The correct answer is:
[tex]\[ \boxed{7 \text{ units}} \][/tex]
