In a triangle ABC , the vertex A is (2, 6) . The internal angle bisectors of angles B and C are x - y + 1 = 0 and x + 3y - 15 = 0 respectively. The area of the region enclosed by the lines AB , AC , and the x -axis is

Step-by-step solution

The image of vertex A with respect to the internal angle bisectors of B and C will lie on the line BC . Let A₁(x₁, y₁) be the image of A(2, 6) in the bisector of B ( x - y + 1 = 0 ): x₁ - 2 1 = y₁ - 6 -1 = -2(2 - 6 + 1) 1^2 + (-1)^2 = 3 x₁ = 5, y₁ = 3 . So, A₁ is (5, 3) . Let A₂(x₂, y₂) be the image of A(2, 6) in the bisector of C ( x + 3y - 15 = 0 ): x₂ - 2 1 = y₂ - 6 3 = -2(2 + 18 - 15) 1^2 + 3^2 = -1 x₂ = 1, y₂ = 3 . So, A₂ is (1, 3) . The line BC passes through A₁ and A₂ . Thus, the equation of BC is y = 3 . Ve