To find the value of [tex]\( h \)[/tex], the height of the wall, let's break down the steps:
1. Identify the right triangle:
The problem states that we are dealing with a [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle. In this type of triangle, the legs are equal in length, and the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg.
2. Given length of one leg:
We are given that the length of one leg of this triangle is [tex]\(6.5\)[/tex] feet.
3. Apply the [tex]$45^{\circ}-45^{\circ}-90^{\circ}$[/tex] triangle theorem:
According to the theorem, if the leg length is [tex]\(6.5\)[/tex] feet, the hypotenuse [tex]\(h\)[/tex] can be calculated as:
[tex]\[
h = 6.5 \times \sqrt{2}
\][/tex]
4. Calculate the hypotenuse [tex]\(h\)[/tex]:
Using the given lengths, the hypotenuse (or height [tex]\(h\)[/tex]) is:
[tex]\[
h \approx 6.5 \times 1.4142 \approx 9.1924
\][/tex]
So, the value of [tex]\( h \)[/tex] is approximately [tex]\(9.1924\)[/tex] feet. Based on the given options, the closest value is [tex]\(6.5 \sqrt{2} \text{ ft}\)[/tex] since [tex]\(6.5 \sqrt{2} \approx 9.1924\)[/tex].
Thus,
[tex]\[
\boxed{6.5 \sqrt{2} ft}
\][/tex] is the value of [tex]\( h \)[/tex], the height of the wall.
