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Question: How do you find the midpoint of the line segment joining \((2, - 4)\) and \((5, - 2)\)?...

How do you find the midpoint of the line segment joining (2,−4)(2, - 4) and (5,−2)(5, - 2)?

Explanation

Solution

We are given a question and in that we have to find out the midpoint of the line segment joined by the two points. When we are given the endpoints of a line segment we can find out its midpoint by using the midpoint formula. As the name might have already suggested, midpoint is basically the halfway between two endpoints. All you need to do is divide the sum of x-values and the sum of y-values by 22.

Complete step-by-step answer:
(X=x1+x22);(Y=y1+y22)\left( {X = \dfrac{{{x_1} + {x_2}}}{2}} \right);\left( {Y = \dfrac{{{y_1} + {y_2}}}{2}} \right)
Herex1{x_1}= x coordinate of one point;x2{x_2}= x coordinate of second point.
y1{y_1}= y coordinate of one point;y2{y_2}= y coordinate of second point. Here X,YX,Y are x and y coordinates of the midpoint.
By using this formula we will find the midpoint.
Step by step answer:
Step1: We are given the coordinate of two points. Let these points be A(2,−4)(2, - 4) and B(5,−2)(5, - 2). We will find the midpoint by using the formula.
(X=x1+x22);(Y=y1+y22)\left( {X = \dfrac{{{x_1} + {x_2}}}{2}} \right);\left( {Y = \dfrac{{{y_1} + {y_2}}}{2}} \right)
Herex1=2{x_1} = 2;x2=5{x_2} = 5;y1=−4;y2=−2{y_1} = - 4;{y_2} = - 2
Step2: For x-coordinate we will substitute the value of x- coordinates in the formula of point A and B.
⇒X=5+22\Rightarrow X = \dfrac{{5 + 2}}{2}
On adding the number and divide it by 22,
⇒X=72\Rightarrow X = \dfrac{7}{2}
On converting it into decimal form as it is improper fraction:
⇒X=3.5\Rightarrow X = 3.5
Step3: For y-coordinate we will substitute the value of y-coordinates in the formula of point A and B.
⇒Y=−4+(−2)2\Rightarrow Y = \dfrac{{ - 4 + \left( { - 2} \right)}}{2}
On adding the number we will get:
⇒Y=−62\Rightarrow Y = \dfrac{{ - 6}}{2}
On converting it into the number we will get:
⇒Y=−3\Rightarrow Y = - 3
So coordinates of the midpoint are (3.5,−3)(3.5, - 3)

Final answer: Hence the midpoint is (3.5,−3)(3.5, - 3).

Note:
In such types of questions students should remember the formula correctly. Students mainly apply the wrong formula. Calculation is short. In case of coordinates in improper fractions convert it into the decimal form. Another word for a point that cuts a line into two equal segments is called the bisector. Some questions may ask you to find the bisector of a line which is basically asking you for a midpoint you may also come across questions asking if a certain coordinate is a bisector and you will have to determine them also with the midpoint formula whether you get the midpoint that was stated if not then it isn't a bisector
Commit to memory:
(X=x1+x22);(Y=y1+y22)\left( {X = \dfrac{{{x_1} + {x_2}}}{2}} \right);\left( {Y = \dfrac{{{y_1} + {y_2}}}{2}} \right)