all lie on a plane

[sanju09]

Points A and B are 6 inches apart. Point M is the midpoint of AB. Point C is 4 inches from point M and point D is 2 inches from point B. Points A, B, C, D, and M all lie on a plane. Which of the following could be the length of CD in inches?

I. 0
II. 1
III. 9

(A) none
(B) I, only
(C) II, only
(D) I and II, only
(E) I, II, and III



https://www.crushthetest.com

[mathbyvemuri]

Let us write all inputs at one place:
AB is a line with M as mid point
AB = 6 and hence AM = BM = 3
CM = 4 and BD = 2

Let us first see what could be the maximum distance between C and D.
"The maximum possible distance between C and D" case arises when all the points C,A,M,B,D lie on the same line in that order.
In that case, CD = CA+AM+MB+BD = CM+MB+BD = 4+3+2 = 9. This means the maximum distance between C and D is 9 and hence statement-III is a valid statement.

To check for distance CD=0 case:
This means both C and D coincide. This can be checked by verifying for the possibility of forming a triangle BCM (or BDM as both the triangles are same when C and D points coincide)
Now let us check the lengths of the sides of supposed triangle BCM
BC = BD = 2; CM = 4; BM = 3;
A triangle with sides 2,3,4 is quite possible (Triangle Inequality Theorem can be checked)
and so the coincidence of C and D. Hence statement-I is also a valid one.

As the statements I and III are valid, all other answer options except "E" can be eliminated.
(If not, by little observation, we can easily say that CD=1 is also quite possible)