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 .

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.