To solve for the length of one leg of a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle given that the hypotenuse measures [tex]\(7 \sqrt{2}\)[/tex] units, we can take advantage of the properties of such triangles. In a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle, the legs are congruent, and each leg is [tex]\( \frac{1}{\sqrt{2}} \)[/tex] times the length of the hypotenuse.
The process is as follows:
1. Identify the relationship in a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle: The hypotenuse [tex]\( c \)[/tex] is related to the legs [tex]\( a \)[/tex] and [tex]\( b \)[/tex] (which are of the same length) by the formula:
[tex]\[
c = a \sqrt{2}
\][/tex]
2. Given hypotenuse: [tex]\( c = 7 \sqrt{2} \)[/tex] units.
3. Set up the equation: Substitute [tex]\( c \)[/tex] into the relationship:
[tex]\[
7 \sqrt{2} = a \sqrt{2}
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
- Divide both sides by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
a = \frac{7 \sqrt{2}}{\sqrt{2}}
\][/tex]
- Simplify the right-hand side:
[tex]\[
a = \frac{7 \sqrt{2}}{\sqrt{2}} = 7
\][/tex]
Hence, the length of one leg of the triangle is [tex]\( \boxed{7} \)[/tex] units.
