Midpoint refers to a point that is in the middle of the line joining two points. The two reference points are the endpoints of a line, and the midpoint is lying in between the two points. The midpoint divides the line joining these two points into two equal halves. Further, if a line is drawn to bisect the line joining these two points, the line passes through the midpoint.
The midpoint formula is used to find the midpoint between two points whose coordinates are known to us. The midpoint formula is also used to find the coordinates of the endpoint if we know the coordinates of the other endpoint and the midpoint. In the coordinate plane, if a line is drawn to connect two points (4, 2), and (8, 6), then the coordinates of the midpoint of the line joining these two points are ({4 + 8}/2, {2 + 6}/2) = (12/2, 8/2) = (6, 4). Let us learn more about the formula of the midpoint, and different methods to find the midpoint of a line.
A midpoint is a point lying between two points and is in the middle of the line joining the two points. If a line is drawn joining the two points, then the midpoint is a point at the middle of the line and is equidistant from the two points. Given any two points, say A and C, the midpoint is a point B which is located halfway between points A and C. Therefore, to calculate the midpoint, we can simply measure the length of the line segment and divide by 2.
Observe that point B is equidistant from A and C. A midpoint exists only for a line segment. A line or a ray cannot have a midpoint because a line is indefinite in both directions and a ray has only one end and thus can be extended.
The midpoint formula is defined for the points in the coordinate axes. Let (x)_{1}, (y)_{1} and (x)_{2}, (y)_{2} be the endpoints of a line segment. The midpoint is equal to half of the sum of the xcoordinates of the two points, and half of the sum of the ycoordinates of the two points. The midpoint formula to calculate the midpoint of a line segment joining these points can be given as,
Given two points A (x)_{1}, (y)_{1} and B (x)_{2}, (y)_{2}, the midpoint between A and B is given by,
M(x)_{3}, (y)_{3} = [(x)_{1} + (x)_{2}]/2, [(y)_{1} + (y)_{2}]/2
where, M is the midpoint between A and B, and (x)_{3}, (y)_{3 }are its coordinates.
Let us look at this example and find the midpoint of two points in onedimensional axis. Suppose, we have two points, 5 and 9, on a number line. The midpoint will be calculated as: (5 + 9)/2 = 14/2 = 7. So, 7 is the midpoint of 5 and 9.
Let's consider a line segment with its endpoints, (x)_{1}, (y)_{1} and (x)_{2}, (y)_{2}. For any line segment, the midpoint is halfway between its two endpoints. The expression for the xcoordinate of the midpoint is [(x)_{1} + (x)_{2}]/2, which is the average of the xcoordinates. Similarly, the expression for the ycoordinate is [(y)_{1} + (y)_{2}]/2, which is the average of the ycoordinates.
Thus, the formula for midpoint is, [(x)_{1} + (x)_{2}]/2, [(y)_{1 }+ (y)_{2}]/2
Let us check an example to see the application of the midpoint formula.
Example: Using the midpoint formula, find the midpoint between points X(5, 3) and Y(7, 8).
Solution: Let M be the midpoint between X and Y.
M = ((5 + 7)/2, (3 + 8)/2) = (6, 11/2)
Therefore, the coordinates of the midpoint between X and Y is (6, 11/2).
Further, based on the points and their coordinate values, the following two methods are used to find the midpoint of the line joining the two points.
Method 1: If the line segment is vertical or horizontal, then dividing the length by 2 and counting that value from any of the endpoints will get us the midpoint of the line segment. Look at the figure shown below. The coordinates of points A and B are (3, 2) and (1, 2) respectively. The length of horizontal line AB is 4 units. Half of this length is 2 units. Moving 2 units from the point (3, 2) will give (1, 2). So, (1, 2) is the midpoint of A B .
Method 2: The other way to find the midpoint is by using the midpoint formula. The coordinates of points A and B are (3, 3) and (1, 4) respectively. Using midpoint formula, we have: ({3 + 1}/2, {3 + 4}/2) = (2/2, 1/2)= (1,1/2).
Method 3: One way to find the midpoint of a line given in a plane is using construction. We can use a compass and straightedge construction to first construct a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point of intersection of the line connecting the cusps and the segment is the midpoint of the segment.
Here's an example to find the coordinate of an endpoint, given the midpoint and the coordinates of the other endpoint.
Example: Midpoint R between the points P and Q has the coordinates (4, 6). If the coordinates of Q are (8, 10), then what are the coordinates for point P? Solve it by using the midpoint formula.
Solution: Let x coordinate of P be m and y coordinate of P be n.
P = (m, n)
Q = (8, 10)
R = (4, 6)
By using the midpoint formula,
R = ((m + 8)/2, (n + 10)/2) = (4, 6)
Solving for m,
(m + 8)/2 = 4
m + 8 = 8
m = 0
Solving for n,
(n + 10)/2 = 6
n + 10 = 12
n = 2
Therefore, coordinates of P are (0, 2).
Important Notes on Midpoint:
The following points are the important properties of the midpoints.
 The midpoint divides a line segment in an equal ratio, that is, 1:1.
 The midpoint divides a line segment into two equal parts.
 The bisector of a line segment cuts it at its midpoint.
The midpoint formula includes computations separately for the xcoordinate of the points, and the ycoordinate of the points. Further, the computations of points between two given points also include similar computation of the xcoordinate and the ycoordinate of the given points. The following two formulas are closely related to the midpoint formula.
The point of intersection of the medians of a triangle is called the centroid of the triangle. The median is a line joining the vertex to the midpoint of the opposite side of the triangle. The centroid divides the median of the triangle in the ratio 2:1. For a triangle with vertices (x)_{1}, (y)_{1}, (x)_{2}, (y)_{2}, (x)_{3}, (y)_{3} the formula to find the coordinates of the centroid of the triangle is as follows.
Section Formula
The section formula is helpful to find the coordinates of any point which is on the line joining the two points. Further, the ratio in which the point divided the line joining the two given points is needed to know the coordinates of the point. The point can be located between the points, or anywhere beyond the points, but on the same line. The section formula to find the coordinates of a point, which divides the line joining the points (x)_{1}, (y)_{1}, and (x)_{2}, (y)_{2} in the ratio m:n is as follows. The positive sign is used in the formula to find the coordinates of the point, which divides the points internally, and the negative sign is used if the point is dividing externally.
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