Introduction to Three Dimensional Geometry · Question 1

Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, -1) and C = (2, 0, -2). Then among the statements (S1) : ABC is an isosceles right angled triangle, and (S2) : the area of ABC is 9 22,

1. both are false
2. only (S2) is true
3. only (S1) is true
4. both are true

Answer

(3) only (S1) is true

Solution

A lies in the xy-plane A = (x, y, 0).

Equidistant from (0, 3, 2) and (0, 0, 1):

x2 + (y - 3)2 + 4 = x2 + y2 + 1 -6y + 12 = 0 y = 2

Equidistant from (2, 0, 3) and (0, 0, 1):

(x - 2)2 + y2 + 9 = x2 + y2 + 1 -4x + 12 = 0 x = 3

A = (3, 2, 0)

AB = (-2, 2, -1), | AB| = 3

AC = (-1, -2, -2), | AC| = 3

AB AC = 2 - 4 + 2 = 0 A = 90

ABC is isosceles right angled: (S1) is true.

Area = 12 3 3 = 92 9 22: (S2) is false.

Answer: (3)