To solve for the height [tex]\( h \)[/tex] of the wall using the [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle theorem, we need to understand the properties of this special type of right triangle.
In a [tex]\( 45^\circ-45^\circ-90^\circ \)[/tex] triangle, the legs are of equal length, and the hypotenuse is [tex]\( \sqrt{2} \)[/tex] times the length of each leg. Let's denote the length of each leg by [tex]\( x \)[/tex].
Given the hypotenuse of the triangle is [tex]\( 6.5 \times \sqrt{2} \)[/tex] feet, we can use this relationship to find [tex]\( x \)[/tex], which represents the height of the triangle (and wall):
[tex]\[ \text{Hypotenuse} = x \times \sqrt{2} \][/tex]
Given:
[tex]\[ 6.5 \times \sqrt{2} = x \times \sqrt{2} \][/tex]
By equating the expressions for the hypotenuse, we can solve for [tex]\( x \)[/tex]:
[tex]\[ x \times \sqrt{2} = 6.5 \times \sqrt{2} \][/tex]
Since both sides have [tex]\( \sqrt{2} \)[/tex], we can divide by [tex]\( \sqrt{2} \)[/tex]:
[tex]\[ x = 6.5 \][/tex]
Therefore, the height [tex]\( h \)[/tex] of the wall is:
[tex]\[ h = 6.5 \text{ feet} \][/tex]
