ABC is a triangle right-angled at B . If A(k, 1, -1) , B(2k, 0, 2) and C(2+2k, k, 1) are the vertices of the triangle, then what is the value of k ?

Step-by-step solution

The direction ratios of the line segment AB are (k - 2k, 1 - 0, -1 - 2) = (-k, 1, -3) . The direction ratios of the line segment BC are (2+2k - 2k, k - 0, 1 - 2) = (2, k, -1) . Since the triangle is right-angled at B , the line segments AB and BC are perpendicular to each other. Therefore, the dot product of their direction ratios is zero. (-k)(2) + (1)(k) + (-3)(-1) = 0 -2k + k + 3 = 0 k = 3 Answer: 3