What is the value of x in the diagram?
The image is of a regular triangle having left, right and base side length of 6, 9 and 10 respectively. Angle at the top vertex is equally bisected by a segment which ends at a point on the base, distance of the point from right side is x.
A.10
B.4
C.8
D.6

 Dear you haven't posted image with question. 

And as per image you HAVEN'T give one dimension BC.. I'm showing you property used in this type of question and how to apply it. After this you can find answer by youself. 

Solution:

Property of angle bisector,

If a ray bisects an angle of a triangle, then it divides the opposite side of the triangle into segments that are proportional to the other two sides. 

AB/BC = AD/DC 

6/BC = 9/DC 

DC = 9BC/6 

See the image and find the answer

OR

Area of FDG: S1 = 1/2 * sin(FDG) * DG * DF

Area of EDG: S2 = 1/2 * sin(EDG) * DG * DE

S1/S2 = 1/2 * sin(FDG) * DG * DF / 1/2 * sin(EDG) * DG * DE

angle EDG = angle FDG by the condition of bisection

S1/S2 = DF/DE

1st answer is incorrect due to ASA, because EGD = 180-FGD, and unless it right triangle it is incorrect

2nd answer is same as 1st and must be correct under the above condition (of 1st answer)

3rd is correct answer

4th is incorrect answer cause they are similar by SAS similarity unless we have 2 side similarity and the angle equality.DE=EG, DF = FG not means DE = or similar to DF

 (DE+DG + angleEDG : DG + DF + angleFDG)