Question

How do you find the midpoint for the line segment joining $A\left( -1,5 \right),B\left( 6,-3 \right)$ ?

Answer 1

We need to find the midpoint of the line segment whose ends are given by the coordinates, $A\left( -1,5 \right),B\left( 6,-3 \right)$ .The point which divides the line segment in exactly half is called the midpoint of that line. Use the formula for finding the midpoint and then evaluate. The obtained coordinate is the midpoint of the given line segment.

Complete step by step solution:
A line segment is a line that is formed by two points. It is of a certain length and has those two points as the endpoints or the corner points.
Now we are given that there is a line segment that is formed by the coordinates $A\left( -1,5 \right),B\left( 6,-3 \right)$ as the endpoints.
We are asked to find the midpoint for this line segment.
A midpoint is a point that divides any line into exactly two equal halves.
It is hence given by the formula,
If there is a line segment formed by the coordinates, $\left( {{x}_{1}},{{y}_{1}} \right),\left( {{x}_{2}},{{y}_{2}} \right)$ then the midpoint of this line is given by, $P\left( x,y \right)=\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)$
Here from the information given in the question,
$\Rightarrow \left( {{x}_{1}},{{y}_{1}} \right)=\left( -1,5 \right)$
$\Rightarrow \left( {{x}_{2}},{{y}_{2}} \right)=\left( 6,-3 \right)$
Upon substituting in the midpoint formula, we get,
$\Rightarrow P\left( x,y \right)=\left( \dfrac{-1+6}{2},\dfrac{5+\left( -3 \right)}{2} \right)$
Upon evaluating we get,
$\Rightarrow P\left( x,y \right)=\left( \dfrac{5}{2},\dfrac{2}{2} \right)$
$\Rightarrow P\left( x,y \right)=\left( \dfrac{5}{2},1 \right)$
Hence, the midpoint for a line segment joining the two points $A\left( -1,5 \right),B\left( 6,-3 \right)$ is given by the coordinate $\left( \dfrac{5}{2},1 \right)$

Note: The midpoint obtained is always equidistant from the given two coordinates. Sometimes we are provided with a ratio that divides the line segment in that ratio. Here in our case, the ratio is equal to 1 since the point is equidistant from both ends. If the ratio is not equal to 1 then there is a different formula to find the point.