Question

A triangle has vertices A(-1,3,4) , B(3,-1,1,) and C(5,1,1). Prove that it is right-angled?

And how do you determine the fourth vertex needed to complete a triangle?
Please help.

Answer 1

`A(-1,3,4) , B(3,-1,1,),C(5,1,1) \\ \\d(A,B)= √((-1-3)^2+(3-(-1))^2+(4-1)^2)= √(16+16+9)= √(41) \\ \\d(B,C)= √((3-5)^2+(-1-1)^2+(1-1)^2)= √(4+4+0)= √(8) \\ \\d(A,C)= √((-1-5)^2+(3-1)^2+(4-1)^2)= √(36+4+9) = √(49) \\ \\AC^2=AB^2+BC^2 \\ \\ √(49)^2=√(41)^2+√(8)^2 \\ \\49 =41+8 \\ \\49=49`

ABC is a right triangle.