Finding the coordinates of the middle of a segment

In this publication, we will consider what the midpoint of a segment is, by what formula its coordinates are calculated (in the plane and in space). We will also analyze examples of solving problems on this topic.

Calculation of the coordinates of the middle of the segment

middle A point is called a point that lies on a segment and is at the same distance from its ends.

AC = CB

If the ends of the segment A (x_ay_a) и B (x_by_b) located in the same plane, then the coordinates of its middle (points C) are calculated according to the formula:

If a segment with ends A (x_ay_a, of_a) и B (x_by_b, of_b) located in three-dimensional space, the coordinates of its middle are calculated as follows:

Examples of tasks

Task 1

Calculate the coordinates of the point C, which is the midpoint of the segment AB formed by the points A (5, -2) и B (11, 10).

Decision:

In this case, the formulas for the plane are suitable for us:

x_c = (5 + 11) / 2 = 8

y_c = (-2 + 10) / 2 = 4

Thus point C has coordinates (8, 4).

Task 2

Find the coordinates of the point B, which is one of the ends of the segment AB. The coordinates of the point are known. A (7, 13) and the middle of the segment C (4, -3).

Decision:

The formulas we need can be derived from expressions for calculating the coordinates of the middle of the segment:

x_b = 2x_c – x_a = 2 · 4 – 7 = 1

y_b = 2y_c – Y_a = 2 · (-3) – 13 =-19

Therefore, the coordinates B are (1, -19).