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The equations of two sides of a triangle are x - y = 0 and 3x + y = 8 . If the orthocenter of the triangle is H(-2, 0) , then the area of the triangle is

Options

  1. A12
  2. B1
  3. C6
  4. D14

Correct answer

C. 6

Step-by-step solution

Let the two given sides be AB: x - y = 0 and AC: 3x + y = 8 . Solving these equations simultaneously gives the coordinates of vertex A : x = y 3x + x = 8 4x = 8 x = 2, y = 2 . So, A (2, 2) . The slope of side AB is 1 . The altitude from C is perpendicular to AB , so its slope is -1 . This altitude passes through the orthocenter H(-2, 0) . Its equation is: y - 0 = -1(x + 2) x + y = -2 Vertex C lies on both AC and the altitude from C . Solving 3x + y = 8 and x + y = -2 : Subtracting the second from the first: 2x = 10

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