To solve for the height [tex]\( h \)[/tex] of the wall using the properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, let's proceed step-by-step:
1. Understand the Properties of a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] Triangle:
- In such a triangle, the lengths of the legs are equal.
- The hypotenuse is a leg length multiplied by [tex]\( \sqrt{2} \)[/tex].
2. Given Information:
- The base length of the triangular portion of the wall is 6.5 feet.
3. Identify the Legs and Hypotenuse:
- Since the triangle is a [tex]\(45^\circ-45^\circ-90^\circ\)[/tex] triangle, both legs (which are the base and the height) are equal.
- Therefore, let each leg be [tex]\( x \)[/tex]. Then, the hypotenuse would be [tex]\( x \sqrt{2} \)[/tex].
4. Set up the Equation:
- Given that the base length (which is one of the legs) is 6.5 feet, we have:
[tex]\[
x = 6.5 \text{ feet}
\][/tex]
- We need to find the height, which is also [tex]\( x \)[/tex] in this case.
5. Solve for the Height:
- Since [tex]\( x = 6.5 \)[/tex], the height [tex]\( h \)[/tex] is:
[tex]\[
h = \frac{6.5}{\sqrt{2}}
\][/tex]
6. Simplify the Height:
- Let's rationalize the denominator to simplify:
[tex]\[
h = \frac{6.5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{6.5 \sqrt{2}}{2}
\][/tex]
7. Calculate Numerically:
- Now let’s determine the numerical value of [tex]\( h \)[/tex]:
[tex]\[
h \approx 4.596194077712559 \text{ feet}
\][/tex]
8. Verify the Possible Options:
- Let’s see which of the provided options matches this value:
[tex]\[
6.5 \text{ feet (Nope, too high)}
\][/tex]
[tex]\[
6.5 \sqrt{2} \text{ feet} \approx 9.19238815542512 \text{ feet (Too high)}
\][/tex]
[tex]\[
13 \text{ feet (Way too high)}
\][/tex]
[tex]\[
13 \sqrt{2} \text{ feet} \approx 18.38477631085024 \text{ feet (Nope, way too high)}
\][/tex]
Therefore, the calculated height [tex]\(h \approx 4.596 \text{ feet}\)[/tex] which simplifies from [tex]\(\frac{6.5 \sqrt{2}}{2}\)[/tex] does not correspond directly to any provided option in its simplified form.
Thus, based on the original height calculation, you can conclude:
[tex]\[
\boxed{4.596194077712559 \text{ feet}}
\][/tex]
