Question

How do you find the midpoint of the line segment joining \[A( - 1,5)B(6, - 3)\]?

Answer 1

Hint: We use the coordinates of each point and calculate the midpoint of the line segment formed using the two given points using the formula of the midpoint of two points. Assume the coordinates of the midpoint and then equate them using the formula.
* Midpoint of two points \[(x,y)\] and \[(a,b)\] is given by\[\left( {\dfrac{{x + a}}{2},\dfrac{{y + b}}{2}} \right)\].

Complete step-by-step answer:
We are given the two points \[A( - 1,5)\] and \[B(6, - 3)\].
Let us assume the line segment formed by joining the points A and B is called AB.
Let the midpoint of AB be named as M having coordinates \[M(x,y)\]
Then we can calculate the coordinates of M using the formula of midpoint of two points i.e. midpoint of two points \[(a,b)\] and \[(c,d)\] is given by\[\left( {\dfrac{{a + c}}{2},\dfrac{{b + d}}{2}} \right)\]
On comparing the values of A and B with general points \[(a,b)\] and \[(c,d)\]
We get \[a = - 1,b = 5,c = 6,d = - 3\]
Substitute these values in the formula
\[ \Rightarrow M(x,y) = \left( {\dfrac{{ - 1 + 6}}{2},\dfrac{{5 - 3}}{2}} \right)\]
Calculate the numerators
\[ \Rightarrow M(x,y) = \left( {\dfrac{5}{2},\dfrac{2}{2}} \right)\]
Cancel possible factors or divide numerator by denominator
\[ \Rightarrow M(x,y) = \left( {2.5,1} \right)\]

\[\therefore \]Midpoint of the line segment joining \[A( - 1,5)\] and \[B(6, - 3)\]is \[\left( {2.5,1} \right)\]

Note:
Many students try to plot the points on the Cartesian plane and then use the distance formula to calculate the midpoint of the line segment joining the given points as they assume the coordinates of the midpoint and then equate both the distances which will be complicated and will not give us the solution. Keep in mind we can use the direct formula of midpoint which gives the value of point exactly at the center of two given points.
Also, students can leave their answer in fraction form if they want to, else in such easy calculations we can convert the fraction into decimal form.