From the information given below, find which of the point is between the other two. If the points are not collinear, state so.

i. d (P, R) = 7, d (P, Q) = 10, d (Q, R) = 3

ii. d (R, S) = 8, d (S, T) = 6, d (R, T) = 4

iii. d (A, B) = 16, d (C, A) = 9, d (B, C) = 7

iv. d (L, M) = 11, d (M, N) = 12, d (N, L) = 8

v. d (X, Y) = 15, d (Y, Z) = 7, d (X, Z) = 8

vi. d (D, E) = 5, d (E, F) = 8, d (D, F) = 6

For any three-distinct collinear points P, Q and R, the point Q is said to be in between P and R if

d (P, Q) + d (Q, R) = d (P, R).

(i) Given: d (P, R) = 7, d (P, Q) = 10, d (Q, R) = 3

Now, consider d (P, Q) = 10 ……….(I)

And d (P, R) + d (Q, R) = 7 + 3 = 10 ……….(II)

∴ Using (I) and (II), we have

d (P, R) + d (Q, R) = d (P, Q)

⇒ R lies between P and Q.

(ii) Given: d (R, S) = 8, d (S, T) = 6, d (R, T) = 4

Here, d (R, S) + d (S, T) = 8 + 6 = 14 ≠ 4 = d (R, T)

Also, d (R, S) + d (R, T) = 8 + 4 = 12 ≠ 6 = d (S, T)

And d (S, T) + d (R, T) = 6 + 4 = 10 ≠ 8 = d (R, S)

Hence, the points R, S and T are non-collinear.

(iii) Given: d (A, B) = 16, d (C, A) = 9, d (B, C) = 7

Now, consider d (A, B) = 16 ……….(I)

And d (C, A) + d (B, C) = 9 + 7 = 16 ……….(II)

∴ Using (I) and (II), we have

d (C, A) + d (B, C) = d (A, B)

⇒ C lies between A and B.

(iv) Given: d (L, M) = 11, d (M, N) = 12, d (N, L) = 8

Here, d (L, M) + d (M, N) = 11 + 12 = 23 ≠ 8 = d (N, L)

Also, d (L, M) + d (N, L) = 11 + 8 = 19 ≠ 12 = d (M, N)

And d (M, N) + d (N, L) = 12 + 8 = 20 ≠ 11= d (L, M)

Hence, the points L, M and N are non-collinear.

(v) Given: d (X, Y) = 15, d (Y, Z) = 7, d (X, Z) = 8

Now, consider d (X, Y) = 15 ……….(I)

And d (Y, Z) + d (X, Z) = 7 + 8 = 15 ……….(II)

∴ Using (I) and (II), we have

d (Y, Z) + d (X, Z) = d (X, Y)

⇒ Z lies between X and Y.

(vi) Given: d (D, E) = 5, d (E, F) = 8, d (D, F) = 6

Here, d (D, E) + d (E, F) = 5 + 8 = 13 ≠ 6 = d (D, F)

Also, d (D, E) + d (D, F) = 5 + 6 = 11 ≠ 8 = d (E, F)

And d (E, F) + d (D, F) = 8 + 6 = 14 ≠ 5 = d (D, E)

Hence, the points D, E and F are non-collinear.