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The equations of the sides AB and AC of a triangle ABC are x - y + 1 = 0 and 2x + y - 4 = 0 respectively. Let H(2a, -a) be the orthocenter of the triangle ABC . If the centroid of the triangle ABC lies on the line x - y = 0 , then the value of BC^2 is equal to

Correct answer

225

Step-by-step solution

Let the vertices of the triangle be A, B, and C . The coordinates of A can be found by solving the equations of AB and AC : x - y = -1 2x + y = 4 Adding these equations gives 3x = 3 x = 1 . Then y = 2 . So, A (1, 2) . The altitude from C is perpendicular to AB . The slope of AB is 1 , so the slope of the altitude from C is -1 . This altitude passes through the orthocenter H(2a, -a) , so its equation is: y - (-a) = -1(x - 2a) x + y = a Vertex C is the intersection of this altitude and AC ( 2x + y = 4 ). Subtracting

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