Midpoint Formula

We will discuss here how to use the midpoint formula to find the middle point of a line segment joining the two co-ordinate points.

The coordinates of the midpoint M of a line segment AB with end points A (x$_{1}$, y$_{1}$) and B (y$_{2}$, y$_{2}$) are M ($\frac{x_{1} + x_{2}}{2}$, $\frac{y_{1} + y_{2}}{2}$).

Let M be the midpoint of the line segment joining the points A (x$_{1}$, y$_{1}$) and B (y$_{2}$, y$_{2}$).

Then, M divides AB in the ratio 1 : 1.

So, by the section formula, the coordinates of M are ($\frac{1\cdot x_{2} + 1\cdot x_{1}}{1 + 1}$) i.e., ($\frac{x_{1} + x_{2}}{2}$).

Therefore, the coordinates of the midpoint of AB are ($\frac{x_{1} + x_{2}}{2}$, $\frac{y_{1} + y_{2}}{2}$).

That is the middle point of the line segment joining the points (x$_{1}$, y$_{1}$) and (y$_{2}$, y$_{2}$) has the coordinates ($\frac{x_{1} + x_{2}}{2}$, $\frac{y_{1} + y_{2}}{2}$).

Solved examples on midpoint formula:

1. Find the coordinates of the midpoint of the line segment joining the point A (-5, 4) and B (7, -8).

Solution:

Let M (x, y) be the midpoint of AB. Then, x = $\frac{(-5) + 7}{2}$ = 1 and y = $\frac{4 + (-8)}{2}$ = -2

Therefore, the required middle point is M (1, -2).

2. Let P (6, -3) be the middle point of the line segment AB, where A has the coordinates (-2, 0). Find the coordinate of B.

Solution:

Let the coordinates of B be (m, n). The middle point P on AB has the coordinates ($\frac{(-2) + m}{2}$, $\frac{0 + n}{2}$).

But P has the coordinates (6, -3).

Therefore, $\frac{(-2) + m}{2}$ = 6  and $\frac{0 + n}{2}$ = -3

⟹ -2 + m = 12 and n = -6

⟹ m = 12 + 2 and n = -6

Therefore, the coordinates of B (14, -6)

3. Find the point A’ if the point A (-3, 4) on reflection in the point (1, -1) maps onto the point A’.

Solution:

Let A’ = (x, y). Clearly, (1, -1) is the middle point of AA’.

The middle point of AA’ = ($\frac{x + (-3) }{2}$, $\frac{y + 4}{2}$) = (1, -1).

⟹ $\frac{x - 3}{2}$ = 1  and $\frac{y + 4}{2}$ = -1

⟹ x = 5 and y = -6

Therefore, the coordinate of the point A’ are (5, -6)

● Distance and Section Formulae

• Distance Formula
• Distance Properties in some Geometrical Figures
• Conditions of Collinearity of Three Points
• Problems on Distance Formula
• Distance of a Point from the Origin
• Distance Formula in Geometry
• Section Formula
• Midpoint Formula
• Centroid of a Triangle
• Worksheet on Distance Formula
• Worksheet on Collinearity of Three Points
• Worksheet on Finding the Centroid of a Triangle
• Worksheet on Section Formula

10th Grade Math

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