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Two vertices of a triangle are A (0, 0) and B (6, 0) . If the orthocentre of the triangle is H (3, 2) , then the value of 16R^2 , where R is the circumradius of the triangle, is:

Options

  1. A144
  2. B169
  3. C160
  4. D208

Correct answer

B. 169

Step-by-step solution

Method 1: The side AB lies on the x-axis. The altitude from the third vertex C is perpendicular to AB and passes through the orthocentre H (3, 2) . Since AB is horizontal, the altitude from C is the vertical line x = 3 . Thus, the x-coordinate of C is 3 . Let C be (3, y_c) . The altitude from A (0, 0) passes through H (3, 2) , so its slope is 2 - 0 3 - 0 = 2 3 . This altitude is perpendicular to the side BC . Therefore, the slope of BC is - 3 2 . Using the coordinates of B (6, 0) and C (3, y_c) , the slope of BC is

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