Question

How do you find the coordinates of the midpoint of AC if A is at $\left( -2,-1 \right)$ and C is at $\left( 4,3 \right)$ ?

Answer 1

Hint: We are given coordinate of two point as $\left( -2,-1 \right)$ and $\left( 4,3 \right)$ . we are asked to find the coordinate of the midpoint of the line segment join by these, to find this we will use the section formula $x=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}$ and $y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}$ . We will find ratio ${{m}_{1}}$ and ${{m}_{2}}$ and then find value of x and y, the point $\left( x,y \right)$ are coordinate to the midpoint.

Complete step-by-step answer:
We are given two points as A and C, whose coordinate is $\left( -2,-1 \right)$ and $\left( 4,3 \right)$ respectively. We are asked to find the coordinate of the midpoint.
Before we start working on our problem, we learn about section formula.
Section formula is the formula which builds a relation between the coordinate of the end point of line segment to the point which divides the line segment In the relation ${{m}_{1}}:{{m}_{2}}$ .
Section formula is given as –
$x=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}$ and $y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}$
Where $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are coordinate of line segment while $\left( x,y \right)$ are coordinate of the point which divide the line segment.
Now in our problem we have that the point dividing into half as midpoint in the point which divides the line segment into equal parts.
So, here the ratio is 1:1, so ${{m}_{1}}=1,{{m}_{2}}=1$ .
Now, we also have a point as $A\left( -2,-1 \right)$ and $C\left( 4,3 \right)$ .
So consider $\left( {{x}_{1}},{{y}_{1}} \right)=\left( -2,-1 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 4,3 \right)$ and coordinate of midpoint as $\left( x,y \right)$ .
Now using above value in $x=\dfrac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}$ and $y=\dfrac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}$
We get –
$\begin{align}
  & x=\dfrac{1\times \left( -2 \right)+1\times 4}{1+1} \\
 & =\dfrac{-2+4}{2} \\
\end{align}$
By simplifying, we get –
$x=\dfrac{2}{2}=1$ and
 $\begin{align}
  & y=\dfrac{1\times \left( -1 \right)+1\left( 3 \right)}{1+1} \\
 & =\dfrac{-1+3}{2} \\
\end{align}$
By simplifying, we get –
$y=1$
So, we get midpoint coordinate are $\left( x,y \right)=\left( 1,1 \right)$

Note: There is shortcut to find the midpoint of the two point joining line.
As the ratio of midpoint is defined so, we have midpoint formula as –
$x=\dfrac{{{x}_{1}}+{{x}_{2}}}{2}$ and $y=\dfrac{{{y}_{1}}+{{y}_{2}}}{2}$ .
Now we have $\left( {{x}_{1}},{{y}_{1}} \right)=\left( -2,-1 \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)=\left( 4,3 \right)$ .
So, using these above, we get –
$\begin{align}
  & x=\dfrac{-2+4}{2},y=\dfrac{-1+3}{2} \\
 & x=\dfrac{2}{2}\text{, }y=\dfrac{2}{2} \\
\end{align}$
So, $x=1,y=1$
So, we get coordinate of midpoint as $\left( 1,1 \right)$