Alright, let's go through this step-by-step to determine the length of one leg of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle when the hypotenuse is given as [tex]\(22\sqrt{2}\)[/tex] units.
1. Understand the Properties of the Triangle:
- In a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, the two legs are of equal length.
- The relationship between the legs and the hypotenuse can be described as follows: if each leg's length is [tex]\(a\)[/tex], then the hypotenuse is [tex]\(a\sqrt{2}\)[/tex].
2. Given:
- Hypotenuse = [tex]\(22\sqrt{2}\)[/tex] units
3. Relate Hypotenuse to Legs:
- Since the hypotenuse ([tex]\(c\)[/tex]) in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle is calculated from the leg ([tex]\(a\)[/tex]) using the formula [tex]\(c = a\sqrt{2}\)[/tex], we can set up the relationship:
[tex]\[
22\sqrt{2} = a\sqrt{2}
\][/tex]
4. Solve for [tex]\(a\)[/tex]:
- To isolate [tex]\(a\)[/tex], divide both sides of the equation by [tex]\(\sqrt{2}\)[/tex]:
[tex]\[
a = \frac{22\sqrt{2}}{\sqrt{2}}
\][/tex]
5. Simplify:
- The [tex]\(\sqrt{2}\)[/tex] in the numerator and the denominator cancel each other out:
[tex]\[
a = 22
\][/tex]
6. Conclusion:
- The length of one leg of the triangle is 22 units.
So, the length of one leg of the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, when the hypotenuse is [tex]\( 22\sqrt{2} \)[/tex] units, is [tex]\( \boxed{22} \)[/tex] units.
