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
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