Question

A line parallel to a triangle's side splits AB into lengths of x - 6 and x. The other side, AC, is split into lengths of x + 6 and x + 20. What is the length of AC? A) 36 B) 46 C) 56 D) 66

Answer 1

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 $AD=x-6$ , $DB=x$, $AE=x+6$ and $EC=x+20$

We need to determine the length of AC

The value of x:

By side splitter theorem, we have,

$\frac{AD}{DB}=\frac{AE}{EC}$

Substituting the values, we have,

$\frac{x+6}{x}=\frac{x+6}{x+20}$

Simplifying, we get,

       $(x+6)(x+20)=x(x+6)$

$x^2+20x-6x-120=x^2+6x$

       $x^2+14x-120=x^2+6x$

               $14x-120=6x$

                          $8x=120$

                            $x=15$

Thus, the value of x is 15

Length of AC:

The length of AC is given by

$AC=AE+EC$

$AC=x+6+x+20$

$AC=15+6+15+20$

$AC=56$

Thus, the length of AC is 56

Hence, Option C is the correct answer.