Midpoint of a Line Segment

Given any two points A and B, the line midpoint is point M that is located at halfway between points A and B.

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Observe that point M is equidistant from points A and B.

A line midpoint can only be found in a line segment. A line or ray cannot have a midpoint as the line is indefinite and can be extended indefinitely in both directions whereas a ray has only one end.

Let us now learn what is the midpoint of a line segment?

What is a Line Segment?

A line segment is a portion of a line that joins two different points.

It is the shortest distance between two points with a definite length that can be measured.

A line segment with two ending points XY is written as \[\overline{XY}\].

Define Midpoint of a Line Segment?

A midpoint of a line segment is the point on a segment that bisects the segment into two congruent segments.

The midpoint of a line segment is the point on a segment that is at the same distance or halfway between the two ending points.

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The Midpoint of a Line Segment Formula

Let (a₁, b₁) and (a₂, b₂) be the ending point of the line segment. The midpoint formula of a line segment joining these two points is given as:

Midpoint Formula

Example:

Suppose we have two points 9 and 5 on a number line, the midpoint of a line will be calculated as:

\[\frac{9 + 5}{2} = \frac{14}{2} = 7\]

Let us learn to find the midpoint of a line segment joined by the ending points (-3, 3) and (5, 3).

Let (-3, 3) be the first endpoint, so a₁ = -3 and b₁ = 3. Similarly, Let (5, 3) be the second endpoint, so a₂ = 5 and b₂ = 3. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (\frac{-3 + 5}{2}, \frac{3 + 3}{2}) = (\frac{2}{2}, \frac{6}{2} = (1, 3)\]

Midpoint Theorem

The statement of the midpoint theorem says that the line segment joining midpoints of the two sides of a triangle is parallel to the third side of a triangle and equal to the half of it. Consider the △ABC given below. Let points D and E be the midpoints of AB and AC. Suppose that you join the points D and E.

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The midpoint theorem says that the line DE will be parallel to the BC and equal to exactly half of BC.

How to Find the Midpoint of a Line Segment?

The midpoint of a line segment can be determined using these two different methods. These are:

1. Counting Method.

2. Using the midpoint of a line segment formula.

Counting Method

If the line segment is vertical or horizontal, you can find the midpoint of a line segment by dividing the length of a line segment by 2 and counting that value from either of the two ending points.

Midpoint Formula Method

The midpoint of a line segment that lies diagonally across the coordinate axis can be found using the midpoint formula.

The midpoint (x,y) of the line segments with ending point A (x₁, y₁) and B(x₂, y₂) can be found using the following midpoint formula.

\[(x, y) = (a, b) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})\]

Example:

Find the midpoint of segment AB, where coordinates of point A and B are (-3, 3) (1, 4) respectively.

Solution:

Using the midpoint formula, we get

\[(\frac{-3 + 1}{2}, \frac{-3 + 4}{2}) = (\frac{-2}{2}, \frac{1}{2}) = (-1, \frac{1}{2})\]

Hence, the midpoint of segment AB is (-1, ½).

The Midpoint of a Line Segment Example with Solutions

1. The diameter of a circle given below has two ending points (2, 3) and (-6, 5). Determine the coordinates of the centre of the circle given below.

Solution:

The centre of a circle divides the diameter into two equal parts. Hence, the coordinates of the centre are the midpoints of a circle.

Let (2, 3) be the first endpoint, so a₁ = 2 and  b₁ = 3. Similarly, Let (-6, 5) be the second endpoint, so a₂ = -6 and b₂ = 5. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (\frac{2 + (-6)}{2}, \frac{-3 + 3}{2}) = (\frac{-4}{2}, \frac{2}{2}) = (-2, 1)\]

Hence, the coordinates of the centre of a circle are (-2, 1).

2. If (3, -2) is the midpoint of the line joining the points (1, x) and (5, 7). Find the value of x.

Solution:

Let (1, h) be the first endpoint, so a₁ = 1 and  b₁ = h. Similarly, Let (5, 7) be the second endpoint, so a₂ = 5 and b₂ = 7. Substitute these points in the midpoint formula given below and simplify to get the midpoint of a line segment.

Using the midpoint formula, we get:

\[(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}) = (3, -2)\]

\[(\frac{1 + 5}{2}, \frac{h + 7}{2}) = (3, -2)\]

\[\frac{7 + h}{2} = -2 = 7 + h = -4\]

\[h= -11\]

Hence, the value of h is -11.