Question

If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then ______.

Options

• AP = \[\frac{1}{3}\text{AB}\]

• AP = PB

• PB = \[\frac{1}{3}\text{AB}\]

• AP = \[\frac{1}{2}\text{AB}\]

Answer 1

If the point P(2, 1) lies on the line segment joining points A(4, 2) and B(8, 4), then  `underlinebb(AP = 1/2 AB)`.

Explanation:

Given that, the point P(2, 1) lies on the line segment joining the points A(4, 2) and B(8, 4), which shows in the figure below:

Now, distance between A(4, 2) and P(2, 1),

AP = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`

AP = `sqrt((2 - 4)^2 + (1 -2)^2`

= `sqrt((-2)^2 + (-1)^2`

= `sqrt(4 + 1)`

= `sqrt(5)`

Distance between A(4, 2) and B(8, 4),

AB = `sqrt((8 - 4)^2 + (4 - 2)^2`

= `sqrt((4)^2 + (2)^2`

= `sqrt(16 + 4)`

= `sqrt(20)`

= `2sqrt(5)`

Distance between B(8, 4) and P(2, 1),

BP = `sqrt((8 - 2)^2 + (4 - 1)^2`

= `sqrt(6^2 + 3^2`

= `sqrt(36 + 9)`

= `sqrt(45)`

= `3sqrt(5)`

∴ AB = `2sqrt(5)`

= 2AP

⇒ AP = `"AB"/2`

Hence, required condition is AP = `"AB"/2`