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Area of a Triangle in Coordinate Geometry
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Mid Point Formula in Coordinate Geometry

Last Updated : 14 Jan, 2025
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Mid point formula in coordinate geometry provides a way to find the mid point of a line segment when the coordinates of the starting and ending points ( i.e. (x1, y1) and (x2, y2) )of the line segment is known. The mid point divides the line in two equal halves i.e the ratio of the sections of the line in which the midpoint divides it is 1:1.

Mid Point of Line
Mid Point of Line

Formula of Mid Point of Line

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 y2, the mid-point of the line will be given by (xm, ym).

The formula for the mid-point  (xm, ym) is:

Mid Point Formula
Mid Point Formula

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

Solution:

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 of Mid Point Formula

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)

Derivation of Mid Point Formula
Derivation of Mid Point Formula

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 Formula

There are similar formulas to the midpoint formula, which are as follows:

  • Section Formula
  • Centroid Formula

Section Formula

Section Formula is used to find the coordinate of the point which divides the given line segment in the desired ratio. Let us assume the endpoints of a line segment are A and B with coordinates (x1, y1) and (x2, y2), and P be the point that divides the line segment joining the line AB in m:n. Then coordinate of P is given by:

P(x, y) = [(mx2 + nx1)/(m+n) , (my2 + ny1)/(m+n)]

Centroid Formula

The Centroid formula is used to find the center point of polygons and mathematically for triangles and quadrilaterals is given as follows:

Centroid of a Triangle Formula

The coordinates of the centroid of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) are:

C(x, y) = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3)

Centroid of triangle
Centroid of Triangle

Centroid of a Quadrilateral Formula

The coordinates of the centroid of a quadrilateral with vertices (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are:

C(x, y) = ((x1 + x2 + x3+ x4)/4, (y1 + y2 + y3+ y4)/4)

Centroid of Quadrilateral
Centroid of Quadrilateral

Solved Questions on Mid-Point Formula

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

Let the midpoint be M(xm, ym), 

xm = (x1 + x2) / 2
x1 = 6, x2 = 3

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

ym = (y1 + y2) / 2
y1 = 8, y2 = 1

Thus, ym = (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).
Solution:

Let the midpoint be M(xm, ym) = (-2, 2.5) where,
x1 = -1, xm = -2

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

xm = (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).
Solution: 

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.

Also Read:

  • Distance Formula
  • Coordinate Geometry

Next Article
Area of a Triangle in Coordinate Geometry

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Article Tags :
  • Mathematics
  • School Learning
  • Class 10
  • Coordinate Geometry
  • Maths-Class-10

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