Question

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Answer 1

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 (x1, y1) and B (y2, y2) are M (x1+x22, y1+y22).

Let M be the midpoint of the line segment joining the points A (x1, y1) and B (y2, y2).

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

So, by the section formula, the coordinates of M are (1⋅x2+1⋅x11+1) i.e., (x1+x22).

Therefore, the coordinates of the midpoint of AB are (x1+x22, y1+y22).

That is the middle point of the line segment joining the points (x1, y1) and (y2, y2) has the coordinates (x1+x22, y1+y22).

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 = (−5)+72 = 1 and y = 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 ((−2)+m2, 0+n2).

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

Therefore, (−2)+m2 = 6 and 0+n2 = -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’ = (x+(−3)2, y+42) = (1, -1).

⟹ x−32 = 1 and y+42 = -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