To solve for the height [tex]\( h \)[/tex] in the given problem involving a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle:
1. Understanding the Properties of a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] Triangle:
- In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the two legs are congruent, meaning they have the same length.
- The hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg.
2. Given Information:
- One leg of the triangle measures 6.5 feet.
3. Application of the Theorem:
- Since the smaller legs are congruent and each measures 6.5 feet,
4. Calculate the Hypotenuse [tex]\( h \)[/tex]:
- The hypotenuse (height of the wall in this context) is calculated as:
[tex]\[
h = 6.5 \times \sqrt{2}
\][/tex]
After calculating this multiplication, we find:
[tex]\[
h \approx 9.19238815542512 \, \text{feet}
\][/tex]
So the height of the wall, represented by the hypotenuse in this specific [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, is approximately [tex]\( 9.19238815542512 \)[/tex] feet. Therefore, none of the provided options ([tex]\( 6.5 \)[/tex] ft, [tex]\( 6.5\sqrt{2} \)[/tex] ft, [tex]\( 13 \)[/tex] ft, [tex]\( 13\sqrt{2} \)[/tex] ft) exactly match this height, but you may refer to the idea that [tex]\( 6.5 \sqrt{2} \)[/tex] ft is the closest representation of the calculation involved.
