Certainly! Let's explore the problem step-by-step in detail.
We are given a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle and the length of its hypotenuse is [tex]\(20 \sqrt{5}\, \text{m}\)[/tex].
### Step 1: Understanding a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] Triangle
A [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle is an isosceles right triangle. This means that the two legs are of equal length and the angles opposite these legs are each [tex]\(45^\circ\)[/tex].
### Step 2: Length Relationships in a [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] Triangle
In such a triangle, the length of each leg is equal to the hypotenuse divided by [tex]\(\sqrt{2}\)[/tex]. Mathematically, if the hypotenuse is [tex]\(h\)[/tex], each leg is [tex]\( \frac{h}{\sqrt{2}} \)[/tex].
### Step 3: Substituting the Given Hypotenuse
We are given that the hypotenuse [tex]\(h\)[/tex] is [tex]\(20 \sqrt{5}\, \text{m}\)[/tex].
### Step 4: Calculating the Length of Each Leg
Using the relationship mentioned, the leg length [tex]\(L\)[/tex] can be calculated as:
[tex]\[
L = \frac{20 \sqrt{5}}{\sqrt{2}}
\][/tex]
### Step 5: Simplifying the Expression
Let's simplify the expression:
1. Combine the roots in the denominator:
[tex]\[
L = \frac{20 \sqrt{5}}{\sqrt{2}} = 20 \cdot \frac{\sqrt{5}}{\sqrt{2}}
\][/tex]
2. Simplify the fraction inside the product:
[tex]\[
L = 20 \cdot \sqrt{\frac{5}{2}}
\][/tex]
### Step 6: Result Interpretation
The final numerical values we've ended up with are:
- Hypotenuse: [tex]\( 44.721359549995796\, \text{m} \)[/tex]
- Each Leg: [tex]\( 31.622776601683793\, \text{m} \)[/tex]
Therefore, the hypotenuse is indeed [tex]\(20 \sqrt{5}\, \text{m}\)[/tex] which is approximately [tex]\( 44.7213\, \text{m} \)[/tex], and each leg is approximately [tex]\(31.6228\, \text{m}\)[/tex].
### Conclusion
Given this detailed process, we successfully determined the measures of both the hypotenuse and the legs of the [tex]\(45^\circ - 45^\circ - 90^\circ\)[/tex] triangle.
