Question

How do you find the midpoint of the segment with the endpoints $\left( {\dfrac{2}{3},\dfrac{9}{2}} \right)$ and $\left( {\dfrac{1}{3},\dfrac{{11}}{2}} \right)$?

Answer 1

Hint: In this question, we need to find the midpoint of the line segment. We denote the given points as $({x_1},{y_1})$ and $({x_2},{y_2})$. We make use of the formula to find the midpoint of a line segment when two end points are given. The formula is given by ${\text{M = }}\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$, where ${\text{M}}$ is the midpoint of the line. Substitute the given points in the formula and obtain the midpoint of the line segment.

Complete step-by-step solution:
We have the given the endpoints for the line segment as $\left( {\dfrac{2}{3},\dfrac{9}{2}} \right)$ and $\left( {\dfrac{1}{3},\dfrac{{11}}{2}} \right)$
We are asked to find the midpoint of the line segment with the given endpoints.
Consider the coordinates $\left( {\dfrac{2}{3},\dfrac{9}{2}} \right)$ to be $({x_1},{y_1})$ and the coordinates $\left( {\dfrac{1}{3},\dfrac{{11}}{2}} \right)$ to be $({x_2},{y_2})$.
The formula to find the midpoint from the two endpoints of the line segment is given by,
${\text{M = }}\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$, where ${\text{M}}$ is the midpoint of the line.
We now substitute the values of ${x_1},$ ${y_1},$ ${x_2}$ and ${y_2}$ in the above formula and simplify to obtain the required midpoint.
Substituting the values we get,
$ \Rightarrow {\text{M = }}\left( {\dfrac{{\dfrac{2}{3} + \dfrac{1}{3}}}{2},\dfrac{{\dfrac{9}{2} + \dfrac{{11}}{2}}}{2}} \right)$
Simplifying further we get,
$ \Rightarrow {\text{M = }}\left( {\dfrac{{\dfrac{{2 + 1}}{3}}}{2},\dfrac{{\dfrac{{9 + 11}}{2}}}{2}} \right)$
$ \Rightarrow {\text{M = }}\left( {\dfrac{{\dfrac{3}{3}}}{2},\dfrac{{\dfrac{{20}}{2}}}{2}} \right)$
Simplifying the terms in the numerator, we obtain,
$ \Rightarrow {\text{M = }}\left( {\dfrac{1}{2},\dfrac{{10}}{2}} \right)$
$ \Rightarrow {\text{M = }}\left( {\dfrac{1}{2},5} \right)$
Therefore the midpoint of the line segment with the endpoints $\left( {\dfrac{2}{3},\dfrac{9}{2}} \right)$ and $\left( {\dfrac{1}{3},\dfrac{{11}}{2}} \right)$ is given by ${\text{M = }}\left( {\dfrac{1}{2},5} \right)$.

Note: The midpoint of a line segment is that point which is at the middle of the line segment. The midpoint of a line segment is found by taking the average of the two x coordinates and the two y coordinates.
i.e. ${\text{M = }}\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)$, where ${\text{M}}$ is the midpoint of the line.
The formula of the slope m of the line can also be calculated from the two endpoints of the line segment. The slope represents how much the change in value for one axis will yield in the change for another axis.
The formula is given by, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.