We will discuss here how to use the midpoint formula to find the middle point of a line segment joining the two coordinate 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
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.