Question 806163
Intriangle;ABC, X and Y are points on sides AB and BC respectively such that XY parallel to AC and XY divides triangular region ABC into two parts equal in area. Then AX/AB is equal to

In triangle AXY & ABC

Angle AXY is congruent to ABC ( corresponding angles)
Angle A is common

By AA test of similarity the triangles 
triangle ABC & AXY are similar

Area of Triangle AXY/area of ABC = AX^2/AB^2

But The ratio of areas = 1/2

1/2 = AX^2/AB^2

1/sqrt(2)= AX/AB