Points X, Y, Z are collinear such that d (X, Y) = 17, d (Y, Z) = 8 find d(X, Z).

Given: X, Y and Z are collinear and

d (X, Y) = 17, d (Y, Z) = 8

We know that

If X, Y and Z are three distinct collinear points, then there are three possibilities:

(i) Point Y is between X and Z

(ii) Point Z is between X and Y

(iii) Point X is between Y and Z

Now, let (i) holds true, i.e. Point Y is between X and Z, then

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

⇒ d (X, Z) = 17 + 8 = 25

⇒ d (X, Z) = 25

Next, let (ii) holds true, i.e. Point Z is between X and Y, then

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

⇒ d (X, Z) = d (X, Y) - d (Y, Z) = 17 – 8 = 9

⇒ d (X, Z) = 9

Lastly, let (iii) holds true, i.e. Point X is between Y and Z, then

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

⇒ d (X, Z) = d (Y, Z) – d (X, Y) = 8 – 17 = -9

⇒ d (X, Z) = -9, which is not possible as distance between any two points is a non-negative real number.

Hence, the value of d (X, Z) is either 25 or 9.