On a number line, points A, B and C are such that d (A, C) = 10, d (C, B) = 8 Find d(A, B) considering all possibilities.

We know that the distance between two points is obtained by subtracting the smaller co-ordinate from the larger co-ordinate.

Given: d (A, C) = 10 and d (C, B) = 8

Now, we know that

If A, B and C are three distinct collinear points, then there are three possibilities:

(i) Point B is between A and C

(ii) Point C is between A and B

(iii) Point A is between B and C

Now, let (i) holds true, i.e. Point B is between A and C, then

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

⇒ d (A, B) = d (A, C) – d (B, C) = 10 – 8 = 2

⇒ d (A, B) = 2

Next, let (ii) holds true, i.e. Point C is between A and B, then

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

⇒ d (A, B) = 10 + 8 = 18

⇒ d (A, B) = 18

Lastly, let (iii) holds true, i.e. Point A is between B and C, then

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

⇒ d (A, B) = d (B, C) – d (A, C) = 8 – 10 = -2

⇒ d (A, B) = -2, which is not possible as distance between any two points is a non-negative real number.

Hence, the value of d (A, B) is either 2 or 18.