A rectangle is bounded by the lines x = 0 , x = 6 , y = 0 , and y = 4 . A line L₁ bisects the area of this rectangle and has a slope of 2 . Let the line L₂ be the reflection of L₁ across the line y = x . If P is the point of intersection of L₁ and L₂ , and A and B are the x-intercepts of L₁ and L₂ respectively, then the area of the triangle PAB is :

Step-by-step solution

The center of the rectangle bounded by x = 0 , x = 6 , y = 0 , and y = 4 is the midpoint of its diagonals, which is ( 0+6 2 , 0+4 2 ) = (3, 2) . Since L₁ bisects the area of the rectangle, it must pass through its center (3, 2) . Given that the slope of L₁ is 2 , its equation is: y - 2 = 2(x - 3) 2x - y - 4 = 0 The line L₂ is the reflection of L₁ across the line y = x . The equation of L₂ can be found by interchanging x and y in the equation of L₁ : 2y - x - 4 = 0 x - 2y + 4 = 0 To find the x-intercept A of L₁ , se