In mathematics and statistics, midpoints are frequently used to determine the middle values. The midpoint is that point that cuts the given line segment into two congruent parts. The midpoint formula is very helpful for finding the middle values.

In this post, we will learn about the definition, rules, formula, and methods of midpoint with a lot of examples.

What is the midpoint?

A midpoint is a point that lies at or near the center of any two ordered pairs. The midpoint can be calculated for 2-D (x and y coordinates), 3-D (x, y, and z coordinates), or so on. In two dimensions, there are two ordered pairs or two endpoints are available.

The term endpoint is correlated with midpoint and is used to evaluate the endpoint by taking the initial point and the midpoint coordinates of a line segment. In simple words, the midpoint is stated as a point that is midway among the endpoints in a line segment.

This term is also referred to as a middle value of the line segment. It divides the line segment into two congruent parts. Suppose the endpoints of a line segment are (x1, y1) and (x2, y2), then add the coordinates of x and y and calculate their means to get the midpoints.

The formula of the midpoint

You can use a midpoint formula, to get the exact center point of a line segment or set of paired numbers. The equation of the midpoint is given below.

Midpoint = M = (x1 + x2)/2, (y1 + y2)/2

You can also use the above equation to find the midpoints separately.

Xm = (x1 + x2)/2

Ym = (y1 + y2)/2

In the above equation, M is the midpoint of a line segment, Xm is the x coordinate of the midpoint, and Ym is the y coordinate of the central point. You can use a midpoint finder to evaluate the midpoint according to the above formula.

Similarly, for three dimensions the equation of the midpoint is given below.

Midpoint = M = (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2

Similarly, you can increase the values of the formula for getting the four dimensions, and so on.

How to calculate the midpoint by using a formula?

Midpoint formulas are used to find the central points of a line segment or set of paired numbers easily. Following are a few examples that are solved by using the midpoint formula.

Example 1: For two dimensions

Calculate the midpoint of the given set of pairs, (3, 6) and (-9, 8).

Solution

Step 1: Write the given set of paired numbers equal to x and y terms.

x1 = 3, x2 = -9, y1 = 6, y2 = 8

Step 2: Now take the formula of midpoint for two dimensions.

Midpoint = M = (x1 + x2)/2, (y1 + y2)/2

Step 3: Now calculate the x coordinate of the midpoint.

Xm = (x1 + x2)/2

Xm = (3 + (-9))/2

Xm = (3 – 9)/2

Xm = (– 6)/2

Xm = – 6/2

Xm = – 3

Step 4: Now find the y coordinates of the midpoint.

Ym = (y1 + y2)/2

Ym = (6 + 8)/2

Ym = (14)/2

Ym = 14/2

Ym = 7

Step 5: Write the midpoints of the given set of paired numbers.

(Xm, Ym) = (-3, 7)

Hence, the midpoint of (3, 6) and (-9, 8) is (-3, 7).

You can also use a midpoint calculator to evaluate the midpoint of a line segment. This calculator works according to the formula of the midpoint. Follow the below steps to get the midpoint.

Step 1: First of all, enter the x and y coordinates.

Step 2: Hit the calculate button to get the midpoint coordinates.

Step 3: A step-by-step result will show in a couple of seconds.

Example 2: For three dimensions

Calculate the midpoint of the given set of pairs, (-9, -7, 12) and (-5, 18, 14).

Solution

Step 1: Write the given set of paired numbers equal to x, y and z terms.

x1 = -9, x2 = -5, y1 = -7, y2 = 18, z1 = 12, z2 = 14

Step 2: Now take the formula of midpoint for two dimensions.

Midpoint = M = (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2

Step 3: Now calculate the x coordinate of the midpoint.

Xm = (x1 + x2)/2

Xm = (-9 + (-5))/2

Xm = (-9 – 5)/2

Xm = (– 14)/2

Xm = – 14/2

Xm = – 7

Step 4: Now find the y coordinates of the midpoint.

Ym = (y1 + y2)/2

Ym = (-7 + 18)/2

Ym = (11)/2

Ym = 11/2

Ym = 5.5

Step 5: Evaluate the z coordinate of the midpoint.

Zm = (z1 + z2)/2

Zm = (12 + 14)/2

Zm = (26)/2

Zm = 26/2

Zm = 13

Step 6: Write the midpoints of the given set of paired numbers.

(Xm, Ym, Zm) = (-7, 5.5, 13)

Hence, the midpoint of (-9, -7, 12) and (-5, 18, 14) is (-7, 5.5, 13).

Example 3

Calculate the midpoint of the given set of pairs, (1, 2, 22) and (11, 12, 15).

Solution

Step 1: Write the given set of paired numbers equal to x, y and z terms.

x1 = 1, x2 = 11, y1 = 2, y2 = 12, z1 = 3, z2 = 15

Step 2: Now take the formula of midpoint for two dimensions.

Midpoint = M = (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2

Step 3: Now calculate the x coordinate of the midpoint.

Xm = (x1 + x2)/2

Xm = (1 + 11)/2

Xm = (12)/2

Xm = 12/2

Xm = 6

Step 4: Now find the y coordinates of the midpoint.

Ym = (y1 + y2)/2

Ym = (2 + 12)/2

Ym = (14)/2

Ym = 14/2

Ym = 7

Step 5: Evaluate the z coordinate of the midpoint.

Zm = (z1 + z2)/2

Zm = (3 + 15)/2

Zm = (18)/2

Zm = 18/2

Zm = 9

Step 6: Write the midpoints of the given set of paired numbers.

(Xm, Ym, Zm) = (6, 7, 9)

Hence, the midpoint of (1, 2, 22) and (11, 12, 15) is (6, 7, 9).

Summary

A midpoint is used to find the central point of the given terms. In this post, we have learned about the midpoint and how to calculate the midpoints of 2 dimensions and three dimensions. By following this post, you can easily solve any problem related to the midpoint.

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