Answer
(a) No
(b) Yes
Work Step by Step
(a) First we find the distances between points:
$|AB| = \sqrt{(3 - 2)^2 + (7 - 4)^2 + (-2 - 2)^2} = \sqrt{26}$
$|BC|=\sqrt{(1-3)^2 + (3 - 7)^2 + (3 - (-2)^2} = \sqrt{45} = 3 \sqrt 5$
$|AC| = \sqrt{(1 - 2)^2 + (3 - 4)^2 + (3 - 2)^2} = \sqrt 3$
In order for the points to lie on a straight line, the sum of the two shortest distances must be equal to the longest distance.
Since the sum of any two segments does not equal the third, the three points do not lie on a straight line.
(b) First we find the distances between points:
$|DE| = \sqrt{(1 - 0)^2 + (-2-(-5))^2+ (4 - 5)^2} = \sqrt{11}$
$|EF|=\sqrt{(3-1)^2+(4-(-2))^2+(2-4)^2} = \sqrt{44} = 2\sqrt{11}$
$|DF| = \sqrt{4-0)^2 + (4 - (-5))^2 + (2 - 5)^2} = \sqrt{99} = 3\sqrt{11}$
Since $|DE| + |EF| = |DF|$, the three points lie on a straight line.
