Midpoint of a Line Segment

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What is Midpoint?

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

\[(a, b) = (\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2})\]


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.

FAQ (Frequently Asked Questions)

1. What does the Midpoint Theorem States?

Ans: The statement of the midpoint theorem says that the line segment joining the midpoint of two triangles is parallel to the third side and is congruent to one half of the third side. Whereas, the statement of the converse of midpoint theorems says that the line is drawn through the midpoint of one side of the triangle and parallel to another side bisects the third side.

2. What is the Use of the Midpoint Formula?

Ans: The midpoint formula is used to find the point that is exactly located between two other points in the plane. The midpoint formula of the line segment is widely used in Geometry.


The coordinates of the point (a, b) that are located exactly at the centre of the two points (a1, b1) and (a2, b2) are given by:


a = (a1 + a2/2), b = (b1 + b2/2)


Similarly, the midpoint of a line segment in a three-dimensional space can be calculated as:

a = (a1 + a2/2), b = (b1 + b2/2), c = (c1 + c2/2)

3. What are Congruent Line Segments?

Ans: Congruent line segments are segments that are of equal length. If segment PQ is congruent to RS, we write as:

P̅Q̅ ≅ R̅S̅

In a line segment, there is one single point that bisects the line segment into two congruent line segments. The point is known as the midpoint.