A triangle has vertices A(2, 1) , B(4, 3) and C(a, b) . If the centroid of the image of this triangle, when reflected across the line x + y - 7 = 0 , is the point (5, 4) , then the value of a + b is :

Step-by-step solution

Let the centroid of the original triangle ABC be G(x₁, y₁) . Since the centroid of the image triangle is the image of the centroid of the original triangle, the point G(x₁, y₁) is the reflection of G'(5, 4) across the line x + y - 7 = 0 . Using the image formula: x₁ - 5 1 = y₁ - 4 1 = -2(5 + 4 - 7) 1^2 + 1^2 x₁ - 5 1 = y₁ - 4 1 = -2(2) 2 = -2 This gives x₁ - 5 = -2 x₁ = 3 and y₁ - 4 = -2 y₁ = 2 . So, the centroid of the original triangle is G(3, 2) . Using the centroid formula for ABC : 2 + 4 + a 3 = 3 a + 6 = 9 a