Option C: 56 is the length of AC
Explanation:
Let DE be the line parallel to BC
Let D divides the side AB and E divides the side E
The lengths of the sides are [tex]AD=x-6[/tex] , [tex]DB=x[/tex], [tex]AE=x+6[/tex] and [tex]EC=x+20[/tex]
We need to determine the length of AC
The value of x:
By side splitter theorem, we have,
[tex]\frac{AD}{DB}=\frac{AE}{EC}[/tex]
Substituting the values, we have,
[tex]\frac{x+6}{x}=\frac{x+6}{x+20}[/tex]
Simplifying, we get,
[tex](x+6)(x+20)=x(x+6)[/tex]
[tex]x^2+20x-6x-120=x^2+6x[/tex]
[tex]x^2+14x-120=x^2+6x[/tex]
[tex]14x-120=6x[/tex]
[tex]8x=120[/tex]
[tex]x=15[/tex]
Thus, the value of x is 15
Length of AC:
The length of AC is given by
[tex]AC=AE+EC[/tex]
[tex]AC=x+6+x+20[/tex]
[tex]AC=15+6+15+20[/tex]
[tex]AC=56[/tex]
Thus, the length of AC is 56
Hence, Option C is the correct answer.
