If the point P (2, 1) lies on the line segment joining points A (4, 2) and B (8, 4),then(A) AP = 13 AB (B) AP = PB (C) PB = 13 AB (D) AP = 12 AB

To solve this problem, we first need to calculate the distances between the points A, B, and P.

1. Calculate the distance between A and P (AP):

   The formula to calculate the distance between two points (x1, y1) and (x2, y2) is √[(x2-x1)² + (y2-y1)²].

   So, AP = √[(2-4)² + (1-2)²] = √[(-2)² + (-1)²] = √[4 + 1] = √5.

2. Calculate the distance between P and B (PB):

   PB = √[(8-2)² + (4-1)²] = √[6² + 3²] = √[36 + 9] = √45.

3. Calculate the distance between A and B (AB):

   AB = √[(8-4)² + (4-2)²] = √[4² + 2²] = √[16 + 4] = √20.

Now, we compare these distances to the options given:

(A) AP = 1/3 AB: √5 is not equal to 1/3 of √20.

(B) AP = PB: √5 is not equal to √45.

(C) PB = 1/3 AB: √45 is not equal to 1/3 of √20.

(D) AP = 1/2 AB: √5 is not equal to 1/2 of √20.

So, none of the options (A, B, C, D) are correct.