Midpoint Formula

A midpoint is the exact middle point of a line segment. It divides the line into two equal parts. The midpoint formula is used to find the point exactly halfway between two points on a coordinate plane.

For a line segment AB in Cartesian coordinate where the x-axis coordinate of point A is x₁ and the y-axis coordinate of point A is y₁ and similarly, the x-axis coordinate of point B is x₂ and the y-axis coordinate of point B is y_2, the mid-point of the line will be given by (x_m, y_m).

The formula for the mid-point  (x_m, y_m) is:

Example: Find the coordinates of the midpoint of a line segment whose endpoints are (5, 6) and (-3, 4).

As we know, the midpoint of a line segment is given by the formula:

Midpoint = ((x₁+x₂)/2 , (y₁+y₂)/2)

where (x₁, y₁) and (x₂, y₂) are the coordinates of the endpoints of the line segment.

Midpoint = ((5+(-3))/2, (6+4)/2)

⇒ Midpoint = (2/2, 10/2)

⇒ Midpoint = (1, 5)

Therefore, the coordinates of the midpoint of the line segment are (1, 5).

Derivation

Let P(x₁,y₁) and Q(x₂,y₂) be the two ends of a given line in a coordinate plane, and R(x,y) be the point on that line which divides PQ in the ratio m₁:m₂ such that

PR/RQ = m₁/m₂  . . .(1)

Drawing lines PM, QN, and RL perpendicular on the x-axis and through R draw a straight line parallel to the x-axis to meet MP at S and NQ at T.

Hence from the figure, we can say:

SR = ML = OL - OM = x - x₁ . . . (2)

RT = LN = ON - Ol = x₂ - x . . . (3)

PS = MS - MP = LR - MP = y - y₁ . . . (4)

TQ = NQ - NT = NQ - LR = y₂ - y . . . (5)

Now triangle ∆SPR is similar to triangle ∆TQR.
Therefore,

SR/RT = PR/RQ

By using equations 2, 3, and 1, we know:

x - x₁ / x₂ - x = m₁ / m₂

⇒ m₂x - m₂x₁ = m₁x₂ - m₁x
⇒ m₁x + m₂x = m₁x₂ + m₂x₁
⇒ (m₁ + m₂)x = m₁x₂ + m₂x₁

⇒ x = (m₁x₂ + m₂x₁) / (m₁ + m₂)

Now triangle ∆SPR is similar to triangle ∆TQR,
Therefore,

PS/TQ = PR/RQ

By using equations 4, 5, and 1, we know:

y - y₁ / y₂ - y = m₁ / m₂

⇒ m₂y - m₂y₁ = m₁y₂ - m₁y
⇒ m₁y + m₂y = m₁y₂ + m₂y₁
⇒ (m₁ + m₂)y = m₁y₂ + m₂y₁

⇒ y = (m₁y₂ + m₂y₁) / (m₁ + m₂)

Hence the coordinates of R(x,y) are:

R(x, y) = (m₁x₂ + m₂x₁) / (m₁ + m₂), (m₁y₂ + m₂y₁) / (m₁ + m₂)

As we had to calculate the midpoint, therefore, we keep the values both of m₁ and m₂ as same i.e. 

For the mid-point we know by the definition of mid-point, m₁ = m₂ = 1.
(x, y) = ((1.x₂ + 1.x₁) / (1 + 1), (1.y₂ + 1.y₁) / (1 + 1))

x, y = (x₂ + x₁) / 2, (y₂ + y₁) / 2

Related Articles:

• Distance Formula
• Coordinate Geometry
• Section Formula
• Centroid Formula

Solved Questions

Question 1: What is the mid-point of line segment AB where point A is at (6,8) and point B is (3,1)?

Let the midpoint be M(x_m, y_m), 

x_m = (x₁ + x₂) / 2
x₁ = 6, x₂ = 3

Thus, x_m = (6 + 3) / 2 = 9 / 2 = 4.5

y_m = (y₁ + y₂) / 2
y₁ = 8, y₂ = 1

Thus, y_m = (8 + 1) / 2 = 9 / 2 = 4.5

Hence the midpoint of line AB is (4.5, 4.5).

Question 2: Find the value of p so that (–2, 2.5) is the midpoint between (p, 2) and (–1, 3).

Let the midpoint be M(x_m, y_m) = (-2, 2.5) where,
x₁ = -1, x_m = -2

y-coordinate of the end point is already known as 2, hence we need to find only the x-coordinate

x_m = (x₁ + x₂) / 2
-2 = (-1 + p) / 2
-4 = -1 +
p = -3

Hence other end-point of line is (-3, 2).

Question 3: If the coordinates of the endpoints of a line segment are (3, 4) and (7, 8), find the distance between the midpoint of the line segment and the point (3, 4).

Let A(3, 4) and B(7, 8) be the endpoints of the given line segment, and C is the midpoint of line segment AB.

Then using midpoint formula,
Coordinate of C = ( (3+7)/2 , (4+8)/2 ) = (5, 6)

Using Distance Formula
Distance = √{(x₂ - x₁)² + (y₂ - y₁)²}
⇒ Distance = √{(3 - 5)² + (4 - 6)²}
⇒ Distance =√{(-2)² + (-2)²}
⇒ Distance =√8 = 2√2

Therefore, distance between midpoint of line segment and point (3, 4) is 2√2.