Question

Show that the points \( (2, -1, 1) \), \( (1, -3, -5) \) and \( (3, -4, -4) \) are the vertices of a right-angled triangle.

Hint:

For checking a right-angled triangle, verify if the sum of squares of two sides equals the square of the third side using the Pythagorean theorem.

Answer 1

We calculate the squared distances between the given points: \[ AB^2 = (1 - 2)^2 + (-3 + 1)^2 + (-5 - 1)^2 \] \[ = (-1)^2 + (-2)^2 + (-6)^2 = 1 + 4 + 36 = 41. \] \[ BC^2 = (3 - 1)^2 + (-4 + 3)^2 + (-4 + 5)^2 \] \[ = (2)^2 + (-1)^2 + (1)^2 = 4 + 1 + 1 = 6. \] \[ CA^2 = (3 - 2)^2 + (-4 + 1)^2 + (-4 - 1)^2 \] \[ = (1)^2 + (-3)^2 + (-5)^2 = 1 + 9 + 25 = 35. \] Since \( AB^2 = BC^2 + CA^2 \) (i.e., \( 41 = 6 + 35 \)), the triangle is right-angled at \( B \).