Final answer:
To find the value of x, we need to use the angle bisector theorem. Given that m(DBC) = x + 6 and m(ABD) = 2x - 12, the equation (x + 6)/(2x - 12) = BD/AD can be set up. By simplifying and factoring the equation, we find that x = 12.
Step-by-step explanation:
To find the value of x, we need to use the angle bisector theorem. According to the theorem, the ratio of the lengths of the segments formed by the angle bisector is equal to the ratio of the lengths of the opposite sides of the triangle.
Given that m(DBC) = x + 6 and m(ABD) = 2x - 12, we can set up the following equation:
(x + 6)/(2x - 12) = BD/AD
Since BD bisects ABC, we can say that BD = CD. Therefore, BD = CD = BC/2.
Substituting the values, we get:
(x + 6)/(2x - 12) = (BC/2)/AB
By cross-multiplying:
(x + 6)(AB) = (2x - 12)(BC/2)
Simplifying the equation, we get:
2x^2 - 30x + 72 = 0
Factoring the equation, we get:
(2x - 6)(x - 12) = 0
Therefore, x = 6 or x = 12.
Since 6 is not an option in the given choices, the value of x is 12.