Mid-Points with its Applications and Graphical Characterization

The concept of mid-point is very common in our daily life. Actually, when we say the mid it means the half of any object. In mathematics, the term is used frequently for nth dimension geometry.

Like the mid-point of a line segment, mid-point of the diameter of a circle, mid of the polygon, and other various subjects like physics, chemistry, and geography depending on the concept of the mid-point.

In this article, we’ll learn about the definition of mid-point, geometrically concepts, and applications in daily life.

What is a Mid-Point?

The term mid-point is defined as the point which is equidistance from its endpoints. It is the point at which the bisector of given endpoints. Sometimes it is also called the average of its endpoints.

Let’s have two points A(x1,y1) and B(x2,y2) then the mid-point C  is found by the given as

C(x, y) = ( x1+x22,y1+y22 )

Mid-Point on Real-line 1-D

If we have two values on a real line then we can easily find the mid-point. The mid-point is just the average of given values. Let us we have two values on a real line first value is 5 and the second is 17. Now the mid-point is the average of 5 and 17.

Mid-point= (5+17) /2

Mid-point= 22/2

Mid-point=11 

The graphical view of the example is given below

Mid-Point on Cartesian plane 2-D

If we have two points in the Cartesian plane then we can calculate the mid-point by finding the mid-value of x- coordinate and y- coordinate. Let’s we have two pints are X(x1,y1) and Y(x2,y2) then the mid-point Z(x,y) can be found as

Z(x, y) = ( x1+x22,y1+y22 )

The x coordinate of mid-point Z is mid of the x-coordinates of X and Y.

The y coordinate of mid-point Z is mid of the y-coordinates of X and Y.

Let’s two points are (5, 4) and (17, 12). Then the mid-point is

 Mid-point = ( 5+172,4+122 )

Mid-point = ( 222,162 )

Mid-point = ( 11,8 )

We can also see this mid-point graphically on the Cartesian plane.

Mid-Point on Hyperplane 3-D

A hyperplane is a plane that has three coordinates named x, y, and z. Let A(x1, y1, z1) and 

C(x2, y2, z2) are two points in the hyperplane. Then the mid-point B(x, y, z) is given as

B(x, y, z) = ( x1+x22,y1+y22, z1+z22 )

Let A (2, 2, 2) and C (4, 4, 4) then B(x, y, z) is mid-point can be obtained by

B(x, y, z) = ( 2+42,2+42, 2+42 )

B(x, y, z) = ( 62,62, 62 )

B(x, y, z) = (3, 3, 3)

Mid-Point on Nth dimension space N-D

We can find the mid-point of all dimension space. The general rule is to find the mid-point of n dimension space if B = (x1,x2,x3,…,xn) and C= (y1,y2,y3,…,yn) then the mid-point A is given as

A= (xn+yn) /2  

We can calculate the mid-point geometrically and analytically easily. For a quick calculation, we can use Mid Point Calculator.

Find the endpoint

To find the one endpoint we have a second endpoint and a mid-point. We can easily find it by the mid-point formula. Let us have a mid-point (10, 10) and one endpoint is (10, 6) then let the second endpoint is A(x, y)

A(x, y) = ( x1+x22,y1+y22 )

(10, 10) =( 10+x2,6+y2 )

Comparing the coordinates on both sides 

1st for x coordinate 

10= 10+x2    

10*2 = 10+x

20 =10+x

20-10 =x

10= x

Similarly for y coordinate 

10= 6+y2

10*2 = 6+y

20 = 6+y

20 –6 = y

14 = y

Now put the value of x and y we have the 2nd endpoint A(x, y) is A (10, 14).

Applications of Mid-Point 

We discuss for application in different geometrical Figures.

1. Centre of Circle 

We can use the concept of mid-point in the circular figures. If we have a circle x^2+y^2=4 with the endpoints of diameter A= (-3, 0) and B= (3, 0) then the center of the circle is 

C (x, y) = ( x1+x22,y1+y22 )

C (x, y) = ( -3+32,0+02 )

C (x, y) = (0, 0)

Its means the center of a circle lies at the origin.

1. Mid-Point of a chord

If we have a chord of a circle with its end points. Then we can find the mid-point of the chord by using the mid-point formula.

The line make a 900 between the mid-point of the chord of the circle and the center of the circle. The graphical view of this is shown in the figure below.

1. Find Circumcenter by using Mid-Points

If we have a triangle ABC and then we find the mid-points of each sides of the triangle. The point of intersection of these mid-points M1, M2, and M3 is point O. The O point is called the circumcentre of the circumcircle. We can understand this by the given diagram.

1. Centre of Mass of a Square

The center of mass is a point of any object where all the weight acts on this point. The center of mass of different object have a different calculation method. If we have a square then the center of mass is obtained by taking the mid-point of the diagonal.  In another way, the two diagonals of the square bisect each other. That point is called the center of mass of the square.

Example 1:

Find the mid-point of a=4 and b=12.

Solution:

Step 1. Write the formula of mid-point.

x= (a+b)/2 where x is called.

Step 2. Put the values a and b.

x= (4+12)/2

x= 16/2

x= 8 

So the mid-point is 8.

Example 2:

If A and B are two points and C is the mid-point of AB. A= (-2,-10) and C= (-10, 2) then find the point B.

Solution:

Step 1: Write the formula of the mid-point.

(x, y) = ( x1+x22,y1+y22 )

Step 2: Put the given values in the formula.

(-10, 2)= (-1+x2,-10+y2 )

Step 3: Now compare the coordinates on both sides.

For x-coordinate 

-10 = -1+x2

-10*2= -1+x

-20 = -1+ x

-19= x

For y-coordinate

2 = -10+y2

2*2 = -10 +y

4 = -10 +y

14 =y

Point B= (-19, 14).

Summary:

Now you can find easily the mid-point of any object. You can also understand the physical meaning of the mid-point with the help of shapes and graphs. You can solve all problems of mid-points by following the above examples.