Question

How do you find the midpoint of the line segment with endpoints $(5,12)$and $(7,-5)$

Answer 1

Hint: In this question we will consider the points $(5,12)$ and $(7,-5)$ to be point $a$ and $b$ respectively and then use the midpoint of a line formula which is $M=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$ where $M$ is the midpoint of a line and ${{x}_{1}},{{y}_{1}},{{x}_{2}},{{y}_{2}}$ are the points of the line. we will then plot the line and the midpoint on the graph.

Complete step-by-step solution:
We have the given points for the line segment as $(5,12)$ and $(7,-5)$.
Consider the coordinates $(5,12)$ to be $({{x}_{1}},{{y}_{1}})$ and the coordinates $(7,-5)$ to be $({{x}_{2}},{{y}_{2}})$.
the formula for finding the midpoint from the two endpoints of the line is $M=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$.
On substituting the values of ${{x}_{1}},{{y}_{1}},{{x}_{2}},{{y}_{2}}$ in the formula, we get:
$\Rightarrow M=\left( \dfrac{5+7}{2},\dfrac{12+(-5)}{2} \right)$
On simplifying the values in the numerator of the fractions, we get:
$\Rightarrow M=\left( \dfrac{12}{2},\dfrac{7}{2} \right)$
On simplifying the values, we get:
$\Rightarrow M=\left( 6,3.5 \right)$, which is the required midpoint of the line segment.
It can be plotted on the graph as:

Where points $A$ and $B$ are the endpoints of the line and $C$ is the midpoint of the line.

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.
The formula of the slope 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 $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.