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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 = \[\frac{1}{2}\text{AB}\]
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Solution
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`
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