Sure, let's solve the problem step by step:
1. Understand the properties of 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, meaning it has two equal sides and one hypotenuse.
- The relationship between the legs (which are equal) and the hypotenuse in this triangle can be described by the following fact: if the length of one leg is [tex]\( x \)[/tex], then the hypotenuse will be [tex]\( x\sqrt{2} \)[/tex].
2. Given information:
- The hypotenuse of the triangle is 128 cm.
3. Find the length of the legs:
- Let's denote the length of each leg by [tex]\( L \)[/tex].
- According to the properties of the [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle, we know:
[tex]\[
\text{Hypotenuse} = L \sqrt{2}
\][/tex]
- Substituting the hypotenuse value:
[tex]\[
128 = L \sqrt{2}
\][/tex]
- To isolate [tex]\( L \)[/tex], divide both sides of the equation by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[
L = \frac{128}{\sqrt{2}}
\][/tex]
4. Simplify the expression:
- To make it easier to handle, you can rationalize the denominator if needed, but in this case, we will directly find the decimal value:
[tex]\[
L = \frac{128}{\sqrt{2}}
\][/tex]
5. Calculate the length of each leg:
- Using the given result, we have:
[tex]\[
L \approx 90.50966799187808 \ \text{cm}
\][/tex]
6. Summarize the result:
- The hypotenuse of the [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle is 128 cm.
- The length of each leg of the triangle is approximately 90.51 cm.
So, we can conclude the solution with the following:
- Hypotenuse: 128 cm
- Each leg: approximately 90.51 cm
