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An isosceles triangle ABC with AB = AC has its centroid at G(2, 2) . The equation of the base BC is x + y - 2 = 0 , and the area of the triangle is 24 square units. The value of AB^2 is equal to

Options

  1. A26
  2. B80
  3. C50
  4. D146

Correct answer

C. 50

Step-by-step solution

Let the altitude from A to BC meet BC at D . In an isosceles triangle, the median to the base is also the altitude, so AD BC . The centroid G(2, 2) lies on the median AD and divides it in the ratio 2:1 . The perpendicular distance from G to the base BC ( x + y - 2 = 0 ) is GD . GD = |2 + 2 - 2| 1^2 + 1^2 = 2 2 = 2 Since AG:GD = 2:1 , the total altitude h = AD = 3 GD = 3 2 . The area of ABC is given as 24 square units. 1 2 BC h = 24 1 2 BC 3 2 = 24 BC = 48 3 2 = 8 2 Since AD bisects BC , BD = BC 2 = 4 2 . In right-a

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