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

Question:

If the point P(2,1) lies on the line segment joining points A(4, 2) and 6(8, 4), then

(a) $A P=\frac{1}{3} A B$

(b) $A P=P B$

(c) $P B=\frac{1}{3} A B$

(d) $A P=\frac{1}{2} A B$

 

Solution:

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

Now, distance between $A(4,2)$ and $(2,1), A P=\sqrt{(2-4)^{2}+(1-2)^{2}}$

$=\sqrt{(-2)^{2}+(-1)^{2}}=\sqrt{4+1}=\sqrt{5}$

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

$A B=\sqrt{(8-4)^{2}+(4-2)^{2}}$

$=\sqrt{(4)^{2}+(2)^{2}}=\sqrt{16+4}=\sqrt{20}=2 \sqrt{5}$

Distance between $B(8,4)$ and $P(2,1), B P=\sqrt{(8-2)^{2}+(4-1)^{2}}$

$=\sqrt{6^{2}+3^{2}}=\sqrt{36+9}=\sqrt{45}=3 \sqrt{5}$

$\therefore$ $A B=2 \sqrt{5}=2 A P \Rightarrow A P=\frac{A B}{2}$

Hence, required condition is $A P=\frac{A B}{2}$

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