Question

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

Answer 1

(i) We have, d(P, R) = 7; d(P, Q) = 10; d(Q, R) = 3
Now, d(P, R) + d(Q, R) = 7 + 3
Or, d(P, R) + d(R, Q) = 10
∴ d(P, Q) = d(P, R) + d(Q, R)
Hence, the points P, R and Q are collinear.
The point R is between P and Q i.e., P-R-Q.

(ii) We have, d(R, S) = 8; d(S, T) = 6; d(R, T) = 4
Now, 8 + 6 = 14, so 8 + 6 ≠ 4; 6 + 4 = 10, so 6 + 4 ≠ 8 and 8 + 4 = 12, so 8 + 4 ​≠ 6
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points R, S and T are non-collinear.

(iii) We have, d(A, B) = 16; d(C, A) = 9; d(B, C) = 7
Now, d(C, A) + d(B, C) = 9 + 7
Or, d(A, C) + d(C, B) = 16
∴ d(A, B) = d(A, C) + d(C, B)
Hence, the points A, C and B are collinear.
The point C is between A and B i.e., A-C-B.

(iv) We have, d(L, M) = 11; d(M, N) = 12; d(N, L) = 8
Now, 11 + 12 = 23, so 11 + 12 ≠ 8; 12 + 8 = 20, so 12 + 8 ≠ 11 and 11 + 8 = 19, so 11 + 8 ​≠ 12
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points L, M and N are non-collinear.

(v) We have, d(X, Y) = 15; d(Y, Z) = 7; d(X, Z) = 8
Now, d(X, Z) + d(Y, Z) = 7 + 8
Or, d(X, Z) + d(Z, Y) = 15
∴ d(X, Y) = d(X, Z) + d(Z, Y)
Hence, the points X, Z and Y are collinear.
The point Z is between X and Y i.e., X-Z-Y.

(vi) We have, d(D, E) = 5, d(E, F) = 8, d(D, F) = 6
Now, 5 + 8 = 13, so 5 + 8 ≠ 6; 8 + 6 = 14, so 8 + 6 ≠ 5 and 5 + 6 = 11, so 5 + 6 ​≠ 8
Since, the sum of the distances between two pairs of points is not equal to the distance between the third pair of points, so the given points D, E and F are non-collinear.