Answer
lPQl = $\sqrt 9$
lQRl = $\sqrt 45$
lPRl = $\sqrt 38$
Not right triangle or Isosceles triangle.
Work Step by Step
Distance formula:
lPQl = $\sqrt ((4-2)^2 + (1+1)^2 + (1-0)^2)) $
simplify
$\sqrt ((2)^2 + (2)^2 + (1)^2) $ =
$\sqrt (4 + 4 + 1) $
= $\sqrt 9 $
lQRl = $\sqrt ((4-4)^2 + (-5-1)^2 + (4-1)^2)) $
simplify
$\sqrt 45 $
lPRl = $\sqrt ((4-2)^2 + (-5+1)^2 + (4-0)^2)) $
simplify
$\sqrt 38 $
Isosceles triangle must have 2 sides that are the same length, since they are all different is not isosceles.
Test for right triangle:
A^2 + B^2 = C^2
$\sqrt 9$ ^2 + $\sqrt 38$ ^2 = $\sqrt 45$ ^2 ?
9 + 38 = 45 ?
47 = 45 ?
No! is not a right triangle.
