Question

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

• A. $AP=\frac{1}{3}AB$
• B. $AP=PB$
• C. $PB=\frac{1}{3}AB$
• D. $AP=\frac{1}{2}AB$

Answer 1

The correct option is D

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

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

Now, distance between

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

[$\because$ Distance between two points $(x_1,y_1)$ and $(x_2,y_2),d=\sqrt{(x_2-x_1{)}^2+(x_2-y_1{)}^2}$]

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

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

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

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

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

$=\sqrt{(8-2{)}^2+(4-1{)}^2}$

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

$\therefore AB=2\sqrt{5}=2AP\Rightarrow AP=\frac{AB}{2}$

Hence, required condition is AP $=\frac{AB}{2}$