Let the vertex A of a triangle ABC be (1, 2) , and the mid-point of the side AB be (5, -1) . If the centroid of this triangle is (3, 4) and its circumcenter is ( , ) , then 21( + ) is equal to:

Step-by-step solution

Let the coordinates of vertex B be (x₁, y₁) and vertex C be (x₂, y₂) . Since the mid-point of AB is (5, -1) , we have: 1 + x₁ 2 = 5 x₁ = 9 2 + y₁ 2 = -1 y₁ = -4 Thus, B (9, -4) . Given the centroid of ABC is (3, 4) , we get: 1 + 9 + x₂ 3 = 3 x₂ = -1 2 - 4 + y₂ 3 = 4 y₂ = 14 Thus, C (-1, 14) . The circumcenter ( , ) is the intersection of the perpendicular bisectors of the sides of the triangle. For side AB , the mid-point is (5, -1) and the slope is -4 - 2 9 - 1 = - 3 4 . The slope of its perpendicular bisector is