If the point P (2,1) lies on the line segment | vedclass

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(C) Given that the point $P(2,1)$ lies on the line segment joining the points $A(4,2)$ and $B(8,4)$.First,we calculate the distance between $A(4,2)$ a

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$AP = \frac{1}{2} AB$

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(C) Given that the point $P(2,1)$ lies on the line segment joining the points $A(4,2)$ and $B(8,4)$.First,we calculate the distance between $A(4,2)$ and $P(2,1)$ using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:$AP = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}$.Next,we calculate the distance between $A(4,2)$ and $B(8,4)$:$AB = \sqrt{(8 - 4)^2 + (4 - 2)^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$.Now,we compare $AP$ and $AB$:$AB = 2\sqrt{5} = 2 	imes (\sqrt{5}) = 2 AP$.Therefore,$AP = \frac{1}{2} AB$.Thus,the correct condition is $AP = \frac{1}{2} AB$.

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

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