Certainly! Let's analyze a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, which is a special right triangle. In such a triangle, the sides have unique relationships with each other:
1. The two legs are congruent (i.e., they have the same length).
2. The hypotenuse is [tex]\( \sqrt{2} \)[/tex] times longer than each leg.
Given that the hypotenuse of the triangle measures [tex]\( 7\sqrt{2} \)[/tex] units, we want to find the lengths of the other two sides (the legs).
Here are the steps:
1. Identify the length of the hypotenuse:
[tex]\[
\text{Hypotenuse} = 7\sqrt{2} \text{ units}
\][/tex]
2. Recall the relationship in a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle: the hypotenuse ([tex]\(c\)[/tex]) is [tex]\( \sqrt{2} \)[/tex] times the length of each leg ([tex]\(a\)[/tex]):
[tex]\[
c = a\sqrt{2}
\][/tex]
3. Substitute the given hypotenuse length into the equation to solve for [tex]\(a\)[/tex]:
[tex]\[
7\sqrt{2} = a \sqrt{2}
\][/tex]
4. To find the length of each leg ([tex]\(a\)[/tex]), divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
a = \frac{7\sqrt{2}}{\sqrt{2}}
\][/tex]
5. Simplify the fraction:
[tex]\[
a = 7
\][/tex]
Therefore, the lengths of the legs of the triangle are each:
[tex]\[
\text{Side length} = 7 \text{ units}
\][/tex]
Given our numerical results, we see that the hypotenuse is [tex]\( 9.899494936611665 \)[/tex] and each leg is [tex]\( 7 \)[/tex] units (more precisely [tex]\(6.999999999999999\)[/tex] due to rounding errors in computational results).
Thus, for practical purposes, each leg is approximately:
[tex]\[
7 \text{ units}
\][/tex]
