Question

Find the coordinates of the midpoint of the segment with the given endpoints.

$\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{5}\mathbf{)}$

Short answer

The coordinates of the midpoint of the segment with the given endpoints is$\left(\frac{{\displaystyle 5}}{{\displaystyle 2}},4\right)$.

Step-by-step solution

Step 1. State the concept of ‘midpoint of a segment in 2 dimensional plane’.

Let the two points in 2 dimensional plane be $(x_1,y_1)$and $(x_2,y_2)$.

The midpoint of the segment with endpoints is given as:

$(x,y)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})\dots \left(1\right)$

Where $(x,y)$is the midpoint.

Step 2. Calculate the distance.

The two points which is given are $(2,3)$and $(3,5)$.

Substitute $(2,3)$as $(x_1,y_1)$and $(3,5)$as $(x_2,y_2)$in $\left(1\right)$to calculate the distance.

$\begin{array}{l}(x,y)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{2+3}{2},\frac{3+5}{2}\right)\\ =\left(\frac{5}{2},\frac{8}{2}\right)\\ =\left(\frac{5}{2},4\right)\end{array}$

Step 3. State the conclusion.

Therefore, midpoint of the segment with the endpoints$(2,3)$ and $(3,5)$is $\left(\frac{5}{2},4\right)$.