Answer:
Option C. 8.5 in.
Step-by-step explanation:
see the attached figure with letters to better understand the problem
we know that
The formula of area of triangle is equal to
[tex]A=\frac{1}{2}(b)(h)[/tex]
In this problem
we have
[tex]b=BC=6\ in[/tex]
[tex]h=AD=x\ in[/tex]
substitute
[tex]A=\frac{1}{2}(6)(x)[/tex]
[tex]A=3x\ in^{2}[/tex] ------> equation 1
Remember that
Heron's Formula is a method for calculating the area of a triangle when you know the lengths of all three sides.
Let
a,b,c be the lengths of the sides of a triangle.
The area is given by:
[tex]A=\sqrt{p(p-a)(p-b)(p-c)}[/tex]
where
p is half the perimeter
p=[tex]\frac{a+b+c}{2}[/tex]
we have
[tex]a=9\ in[/tex]
[tex]b=9\ in[/tex]
[tex]c=6\ in[/tex]
Find the half perimeter p
p=[tex]\frac{9+9+6}{2}=12\ in[/tex]
Find the area
[tex]A=\sqrt{12(12-9)(12-9)(12-6)}[/tex]
[tex]A=\sqrt{12(3)(3)(6)}[/tex]
[tex]A=\sqrt{648}[/tex]
[tex]A=25.46\ in^{2}[/tex]
Substitute the value of the area in the equation 1 and solve for x
[tex]A=3x\ in^{2}[/tex]
[tex]25.46=3x[/tex]
[tex]x=25.46/3[/tex]
[tex]x=8.5\ in[/tex]
