Question

What is the midpoint of PQ if $P(-5,-3)$ and $Q(-3,-5)$?

Answer 1

Hint: First we will assume that R is the midpoint of PQ and the coordinates of R will be $\left( x,y \right)$. Then we will use the formula of midpoint which is given as \[R=\left( \dfrac{x{{}_{1}}+x{{}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\], here $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are the coordinates of point P and Q respectively. Then substituting the values and simplifying the obtained equations we will get the desired answer.

Complete step-by-step solution:
We have been given the coordinates of points of a line PQ as $P(-5,-3)$ and $Q(-3,-5)$.
We have to find the coordinates of midpoint of PQ.
Let us assume that the midpoint of PQ be R and the coordinates of R will be $\left( x,y \right)$.
Now, we know that if PQ is a line segment and $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ are the coordinates of points P and Q respectively and R is the midpoint of the line segment PQ then the coordinates of R will be given by the formula \[R\left( x,y \right)=\left( \dfrac{x{{}_{1}}+x{{}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\].
Now, when we compare the coordinates we will get the values
$\Rightarrow {{x}_{1}}=-5,{{y}_{1}}=-3,{{x}_{2}}=-3,{{y}_{2}}=-5$
Now, substituting the values in the formula we will get
\[R\left( x,y \right)=\left( \dfrac{-5-3}{2},\dfrac{-3-5}{2} \right)\]
Now, simplifying the above obtained equation we will get
\[\begin{align}
  & R\left( x,y \right)=\left( \dfrac{-8}{2},\dfrac{-8}{2} \right) \\
 & R\left( x,y \right)=\left( -4,-4 \right) \\
\end{align}\]
Hence we get the coordinates of the midpoint as\[\left( -4,-4 \right)\].

Note: The midpoint of a line segment divides the line in to two equal halves so it is also known as the bisector of a line. If the points are straight then it is easy to calculate the distance between two points on a line. Be careful while substituting the values and solving.