A [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, also known as an isosceles right triangle, has certain properties that stem from its angles. Specifically, the legs of the triangle are congruent, and the hypotenuse is related to the length of each leg by a factor of [tex]\(\sqrt{2}\)[/tex].
Given that the hypotenuse [tex]\(h\)[/tex] is [tex]\(7\sqrt{2}\)[/tex], we need to find the length of each leg [tex]\(a\)[/tex].
The relationship between the hypotenuse and the legs in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle can be given by:
[tex]\[ h = a \sqrt{2} \][/tex]
We can solve for [tex]\(a\)[/tex] by isolating it on one side of the equation:
[tex]\[ a = \frac{h}{\sqrt{2}} \][/tex]
Substitute the given hypotenuse [tex]\(h = 7\sqrt{2}\)[/tex] into the equation:
[tex]\[ a = \frac{7\sqrt{2}}{\sqrt{2}} \][/tex]
When you divide [tex]\(7\sqrt{2}\)[/tex] by [tex]\(\sqrt{2}\)[/tex], the [tex]\(\sqrt{2}\)[/tex] terms cancel out:
[tex]\[ a = \frac{7 \cancel{\sqrt{2}}}{\cancel{\sqrt{2}}} \][/tex]
[tex]\[ a = 7 \][/tex]
Thus, the length of each leg of the triangle is 7.
So the correct answer is:
[tex]\[ \boxed{7} \][/tex]
