To solve for the height [tex]\( h \)[/tex] of the wall, we need to use the properties of a 45°-45°-90° triangle. In such a triangle, the lengths of the legs are equal to each other, and the length of the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg.
Given:
- The hypotenuse of the triangle is 6.5 ft.
Let's determine the height [tex]\( h \)[/tex]:
1. Understanding the relationship:
In a 45°-45°-90° triangle,
[tex]\[
\text{Hypotenuse} = \text{Leg} \times \sqrt{2}
\][/tex]
Here, the hypotenuse is given as 6.5 ft.
2. Finding the length of each leg (which is the height [tex]\( h \)[/tex]):
[tex]\[
h = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{6.5 \text{ ft}}{\sqrt{2}}
\][/tex]
According to the calculations, dividing 6.5 by [tex]\( \sqrt{2} \)[/tex] gives us:
[tex]\[
h \approx 4.596194077712559 \text{ ft}
\][/tex]
3. Listing the choices:
- 6.5 ft
- [tex]\( 6.5 \sqrt{2} \)[/tex] ft
- 13 ft
- [tex]\( 13 \sqrt{2} \)[/tex] ft
From these steps, we see that the height [tex]\( h \)[/tex] of the wall is approximately 4.596 ft, which doesn't exactly match any of the provided options. Therefore, we should consider the correctness of the problem or if additional context might have been misplaced, but based on your numerical results, the height [tex]\( h \approx 4.596 \text{ ft} \)[/tex].
