Midpoint of a Line Segment

Midpoint of a Line Segment is the point which divides the line segment into two equal parts. It lies exactly between the endpoints of the line. The midpoint of a line segment is highly useful for solving various geometrical problems. With the help of the midpoint, we can find the center of an object, line, or any other curve.

In this article, we will learn about line segments, the midpoint of a line segment, the formula to calculate the midpoint of a line segment, and then we will see some practice problems on how to find the midpoint of a line segment.

What is Line Segment?

Line segment is a straight path that does not curve or bend. It is a part of a line that has a definite beginning and end, represented by two distinct endpoints.

A line segment always follows a straight path and has a definite length. The length of a line segment can be calculated by calculating the distance between the endpoints of the line segment using the Distance Formula, which is given below:

Distance = √ (x₂ -x_{1 )} + (y₂ -y₁)

where (x_2, x₁) and (y₂, y₁) represent the coordinate of the endpoints of the line segment.

What is Mid-Point of a Line Segment?

Mid point of a line segment is the point lying in the Mid of line.

It is equidistant from both the ends of the line. Mid point divides the line segment in two equal parts.

Midpoint Formula

Since mid point lies exactly in the Mid of line segment so coordinates of mid point can be calculated using the mid point formula as

• x coordinate of the mid point will be equal to the sum of x coordinate of end points divided by 2.
• y coordinate of the mid point will be equal to the sum of y coordinate of end points divided by 2.

OR

Coordinate of Mid point = ( x₁ + x₂ / 2 , y₁ + y₂ / 2 )

Where

• (x₁ , y₁ ) and (x₂ , y₂) represents the coordinates of end point of the line.

Midpoint Formula for Three Dimension

The mid point formula for two points in three dimension is given below:

Coordinate of Mid point = ( ( x₁ + x₂ ) / 2 , (y₁ + y₂ ) / 3 , (z₁ + z₂ ) / 2 )

where

(x₁ , y₁ , z₁ ) and (x₂ , y₂ , z₂ ) represents the coordinate of end point of the line in 3 Dimensions.

Read More,

• Perpendicular Bisector
• Mid Point Theorem
• Practice Questions on Coordinate Geometry

Solved Examples on Midpoint of a Line Segment

Example 1: Find the Midpoint of line segment joining the points (-3,6 ) and (7,2) ?

Solution :

Given points are (-3, 6) and (7, 2) .

X coordinate of Mid Point = x₁ + x₂ / 2 = ( -3 + 7) / 2 = 4/2 = 2 .

Y coordinate of Mid Point = y₁ + y₂ / 2 = ( 6 + 2 ) / 2 = 8/2 = 4 .

So the coordinate of mid point of given points are (2,4) .

Example 2 : Find the Midpoint of line whose end points are represented as (acos²θ , bsin²θ) and (asin²θ , bcos²θ) ?

Solution :

The given points are (acos²θ , bsin²θ) and (asin²θ , bcos²θ) .

X coordinate of Mid Point = x₁ + x₂ / 2 = ( acos²θ + asin²θ) / 2 = a (cos²θ + sin²θ)/2 = a/2 .

Y coordinate of Mid Point = y₁ + y₂ / 2 = ( bsin²θ + bcos²θ) / 2 = b (sin²θ + cos²θ) /2 = b / 2 .

So the coordinate of mid point of given points are (a/2 , b/2 ) .

Note : Sin²θ + Cos²θ = 1

Example 3 : If the Midpoint of (h ,3) and (12 ,5) is (8 , 4) then find the value of h ?

Solution :

The given Points are (12 , 5 ) and (h,3 ) .

The x coordinate of mid Point is = (12 + h ) / 2

Comparing the x coordinate with the x coordinate of given point we have

(12 + h ) / 2 = 8

or

12 + h = 16

or

h = 4 .

So the required value of h is 4 .

Example 4 : If (h , k ) represents the Midpoint of line segment joining (7,-3) and ( 3, 7) then find the coordinates of p represented as (h²-5k , 2h - 5k ) .

Solution :

Given (h , k ) represents the coordinates of mid point of (7,-3 ) and (3,7 ) .

So ,

h = (sum of x coordinate of given points ) / 2 i.e. (7 + 3 ) / 2 = 5

and

k = (sum of y coordinate of given points )/2 i.e. (-3 + 7 ) 2 = 4/2 = 2 .

So (h , k ) is (5 , 2 ) .

Now

h² - 5k = 5² - 5 . 2 = 25 - 10 = 15

and

2h - 5k = 2.5 - 5.2 = 0

So the required coordinates of p are ( 15 , 0 ) .

Example 5: Find the value of x and y if the Midpoint of line joining A (-3, 5 ) and B ( 7 , 7 ) is M ( x³ -6 , y² - 3 ) .

Solution :

Given M represents the coordinates of Mid point of AB . So ,

Let (h , k ) be the coordinates of M i.e. Mid point of AB

So

h = (sum of x coordinate of given points ) / 2 i.e. (-3 + 7 ) / 2 = 2

and

k = (sum of y coordinate of given points )/2 i.e. (5+ 7 ) 2 = 12/2 = 6 .

Comparing the given coordinates of M and the coordinates of M from Mid Point formula .

x³ - 6 = h = 2

or x³ = 8

or x = 2

Similarly ,

y² - 3 = k = 6

or

y² = 9

and y = 3 or y = -3

So the required value of x is 2 and required value of y are = 3 or -3 .

Practice Question on Mid Point of a line Segment

Q1. Find the Midpoint of line joining ( 2 , - 6 ) and (8 , - 8 ) ?

Q2 . Find the Midpoint of line joining p ( 7 , 1 ) and q , where q represents the Mid point of A ( 8 , -5) and B ( -6 , 3 ) ?

Q3 : If M( 4, -4 ) represents the Mid point of A ( h , k ) and B ( 7 , 9) , find the coordinates of A then ?

Q4 : Find the value of x² - 3x + y , where x and y represents the coordinate of Mid Point of A( 6 , 0 ) ( 0 , 4 ) ?

Q5 : Find the Mid point of A ( sinθ , tanθ ) and B ( cosθ , cotθ) when θ = π / 4 .

Q6 : Find the distance between the point A( 7 , 3 ) and Mid point of AB , where B ( -1, 3) ?

Answer Key

1. ( 5 , -7 )
2. ( 4 , 0 )
3. ( 1 , -17 )
4. 2
5. ( 1/√2 , 1 )
6. 4