In an equilateral triangle PQR , let the vertex P be at (3, 5) and the side QR be along the line x + y = 4 . If the orthocentre of the triangle PQR is ( , ) , then 9( + ) is equal to:

Step-by-step solution

In an equilateral triangle, the orthocentre coincides with the centroid. The centroid divides the altitude from the vertex to the opposite side in the ratio 2:1 . Let M be the foot of the perpendicular from P(3, 5) to the line QR given by x + y - 4 = 0 . The coordinates of M can be found using the formula: x - 3 1 = y - 5 1 = - 3 + 5 - 4 1^2 + 1^2 x - 3 1 = y - 5 1 = -2 This gives x = 1 and y = 3 . Thus, M is (1, 3) . The orthocentre H( , ) divides the segment PM internally in the ratio 2:1 . Using the section form