Question

For the point P(-3,0) and Q(4,3), find the distance d(P,Q) and the coordinates ofthe midpoint M of the segment PQ.

Answer 1

SOLUTION

Write out the given coordinate of point P and Q

$\begin{gathered} P=(-3,0) \\ \text{and} \\ Q=(4,3) \end{gathered}$

The distance between two point is given by

$\text{dist}=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}^{}$

Given point P(-3,0) and Q(4,3)

$\begin{gathered} x_2=4,x_1=-3,y_2=3,y_1=0 \\ \text{Then } \\ \text{dist(P,Q)}=\sqrt[]{(4-(-3)^2+(3-0)^2} \end{gathered}$

Hence, by simplification, we have

$\begin{gathered} \text{dist(P,Q)}=\sqrt[]{7^2+3^2}=\sqrt[]{49+9}=\sqrt[]{58} \\ \end{gathered}$

Hence

The distance between point P and Q is √58 unit

Then

The coordinates of the midpoint is given by

$\begin{gathered} Let\text{ the coordinate of the midpoint m be } \\ (x_m,y_m) \\ \text{Hence } \\ (x_m,y_m)=(\frac{x_1+x_2}{2}+\frac{y_1+y_2}{2}) \end{gathered}$

Where

$\begin{gathered} x_2=4,x_1=-3,y_2=3,y_1=0 \\ \text{Then} \\ (x_m,y_m)=(\frac{-3+4}{2},\frac{3+0}{2}) \\ \\ (x_m,y_m)=(\frac{1}{2},\frac{3}{2}) \end{gathered}$

Therefore

The coordinates of the midpoint M of the segment PQ is (1/2,3/2)