Find the lengths of the sides of the triangle pqr. p(6, −1, 0), q(8, 0, 2), r(9, −2, 0) |pq| = |qr| = |rp| =

PQ = 3 
QR = 3 
RP = sqrt(10) ≠3.16227766  
This is just a matter of using the Pythagorean theorem for each pair of points. Just take the square root of the sum of square of the differences in the X, Y, and Z coordinates. In case, you don't think that the Pythagorean theorem can be used in 3 dimensions, imagine attempting to calculate the length of a diagonal between opposite corners of a 3 dimensional box with height h, depth d, and width w. The length of the diagonal for the top or bottom of the box is sqrt(d^2 + w^2). And then you can calculate the length of the diagonal between the opposite corners by introducing height, which would be sqrt(h^2 + sqrt(d^2 + w^2)^2) and that simplifies to sqrt(h^2 + d^2 + w^2). So with that in mind, let's calculate the lengths: 
PQ = sqrt((6-8)^2 + (-1 - 0)^2 + (0 - 2)^2) 
= sqrt((-2)^2 + (-1)^2 + (-2)^2) 
= sqrt(4 + 1 + 4) 
= sqrt(9) 
= 3  
QR = sqrt((8 - 9)^2 + (0 - -2)^2 + (2 - 0)^2) 
= sqrt((-1)^2 + (2)^2 + (2)^2) 
= sqrt(1 + 4 + 4) 
= sqrt(9) 
= 3 
 RP = sqrt((9 - 6)^2 + (-2 - -1)^2 + (0 - 0)^2) 
= sqrt((3)^2 + (-1)^2 + (0)^2)
 = sqrt(9 + 1 + 0)
 = sqrt(10) 
≠3.16227766