Answer :
To find the length of each leg in a [tex]$45^\circ-45^\circ-90^\circ$[/tex] triangle with a given hypotenuse of [tex]\(7\sqrt{2}\)[/tex], we can take advantage of the properties specific to this type of triangle. In a [tex]$45^\circ-45^\circ-90^\circ$[/tex] triangle, the lengths of the legs are equal, and the relationship between the legs and the hypotenuse is given by the formula:
[tex]\[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} \][/tex]
Given that:
[tex]\[ \text{Hypotenuse} = 7\sqrt{2} \][/tex]
To solve for one leg length, we rearrange the formula:
[tex]\[ \text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Substituting the given hypotenuse:
[tex]\[ \text{Leg} = \frac{7\sqrt{2}}{\sqrt{2}} \][/tex]
Now, simplifying the fraction:
[tex]\[ \text{Leg} = 7 \][/tex]
Therefore, the length of each leg of the triangle is:
[tex]\[ \boxed{7} \][/tex]
[tex]\[ \text{Hypotenuse} = \text{Leg} \times \sqrt{2} \][/tex]
Given that:
[tex]\[ \text{Hypotenuse} = 7\sqrt{2} \][/tex]
To solve for one leg length, we rearrange the formula:
[tex]\[ \text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} \][/tex]
Substituting the given hypotenuse:
[tex]\[ \text{Leg} = \frac{7\sqrt{2}}{\sqrt{2}} \][/tex]
Now, simplifying the fraction:
[tex]\[ \text{Leg} = 7 \][/tex]
Therefore, the length of each leg of the triangle is:
[tex]\[ \boxed{7} \][/tex]
