Question

In $\vartriangle LMN,D$ is the midpoint of the segment $MN$. If $L \equiv \left( {2,4} \right),D \equiv \left( {2, - 2} \right)$then, the coordinates of the point $G$ which divides the median internally in the ratio $2:1$, are:
A) $G\left( {x,y} \right) = \left( {2,2} \right)$
B) $G\left( {x,y} \right) = \left( { - 2,0} \right)$
C) $G\left( {x,y} \right) = \left( { - 2, - 2} \right)$
D) $G\left( {x,y} \right) = \left( {2,0} \right)$

Answer 1

Hint: First of all make a diagram of the question from the given information, then we know the coordinates of $L$ and also of the midpoint $D$.
So, we will find the coordinates of the $G$ by section formula, because as we know that $G$ divides the median in the ratio $2:1$
Section formula states that the coordinates of the point dividing a line segment drawn from point $({x_1},{y_1})$to point $({x_2},{y_2})$in the ratio $m:n$, internally is given by $\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$

Complete step-by-step answer:
Here in this question we will start solving it after enlisting the given information. Now, here in the question it is given that
In the $\vartriangle LMN,D$ is the midpoint of the segment $MN$
Coordinates of the point$L \equiv \left( {2,4} \right)\& D \equiv \left( {2, - 2} \right)$
The point $G$ divides the median internally in the ratio $2:1$
Now, we have to construct a diagram from the given information

So, here from the given information, we can say that we know the coordinates of point $L$ and point $D$ and we also know the ratio in which point $G$ divides the line $LD$(as it the median of the line $MN$)
So, by applying section formula on the line $LD$ we can find the coordinates of point $G$.
So, now using section formula we can say
$G\left( {x,y} \right) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$
Where, $G\left( {x,y} \right)$refers to the $x\& y$ coordinates of point $G$.
$m\& n$ refers to the ratio in which point $G$ divides the line$LD$.
$\left( {{x_1},{y_1}} \right)$refers to $x\& y$ coordinates of point $L$
And $\left( {{x_2},{y_2}} \right)$refers to $x\& y$ coordinates of point $D$.
So, now as it is given that
$\left( {{x_1},{y_1}} \right) = \left( {2,4} \right)$
$\left( {{x_2},{y_2}} \right) = \left( {2, - 2} \right)$
And $m = 2\& n = 1$
So, now putting the corresponding values in the formula
$G\left( {x,y} \right) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)$
$
   \Rightarrow G\left( {x,y} \right) = \left( {\dfrac{{2\left( 2 \right) + 1\left( 2 \right)}}{{2 + 1}},\dfrac{{2\left( { - 2} \right) + 1\left( 4 \right)}}{{2 + 1}}} \right) \\
   \Rightarrow G\left( {x,y} \right) = \left( {\dfrac{{4 + 2}}{3},\dfrac{{4 - 4}}{3}} \right) \\
   \Rightarrow G\left( {x,y} \right) = \left( {\dfrac{6}{3},\dfrac{0}{3}} \right) \\
   \Rightarrow G\left( {x,y} \right) = \left( {2,0} \right) \\
 $

$\therefore $Option D is correct.

Note: In this type of question students generally make mistakes while putting the value of ratio in which line is divided. So, to avoid these types of mistakes, first of all make a diagram and then try to relate what the language of question is asking you to solve. When we divide the points internally then the point lies between the two points. When we divide the points externally then the point lies outside the two points.