Two vertices of a triangle ABC are A(-1, 3) and B(4, -5) . The image of the centroid of the triangle ABC in the line x + y - 4 = 0 is the point P(6, 2) . If the third vertex is C( , ) , then the value of + is equal to:

Step-by-step solution

Let the centroid of the triangle ABC be G(x, y) . Since P(6, 2) is the image of G in the line x + y - 4 = 0 , G is also the image of P in the same line. Using the reflection formula for the point P(6, 2) in the line x + y - 4 = 0 : x - 6 1 = y - 2 1 = -2(6 + 2 - 4) 1^2 + 1^2 x - 6 1 = y - 2 1 = -2(4) 2 = -4 x - 6 = -4 x = 2 y - 2 = -4 y = -2 Thus, the centroid is G(2, -2) . The centroid of ABC with vertices A(-1, 3) , B(4, -5) , and C( , ) is given by: ( -1 + 4 + 3 , 3 - 5 + 3 ) = (2, -2) Equating the coordinates: