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Let ABCD be a square such that its diagonal BD lies on the line x - 2y = 0 and the vertices B and D are symmetric with respect to the origin. If the vertex A lies on the line x + y = 6 and the x-coordinate of D is positive, then the x-intercept of the line containing the side CD is

Options

  1. A30
  2. B10
  3. C-30
  4. D-10

Correct answer

B. 10

Step-by-step solution

Since B and D lie on x - 2y = 0 and are symmetric with respect to the origin, the origin (0,0) is the midpoint of BD and hence the center of the square. The diagonals of a square are perpendicular bisectors of each other. The diagonal AC passes through the origin and is perpendicular to BD . The slope of BD is 1 2 , so the slope of AC is -2 . The equation of AC is y = -2x , or 2x + y = 0 . Vertex A is the intersection of 2x + y = 0 and x + y = 6 . Solving these, we get x = -6 and y = 12 . So, A is (-6, 12) . Since

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