An important concept of coordinate geometry is the midpoint formula. The midpoint of any two known coordinates on a coordinate plane can be easily found by using the midpoint formula. Also, the reverse of the same concept is true, i.e., if the mid-point of any segment and one of the endpoints are known, the coordinates of the other endpoint can be easily known using the relation of the midpoint formula.
- The importance of the formula is assessed in coordinate geometry and graph plotting, where we want to assess the center of the already defined points.
- As the name suggests, the midpoint lies half the way between two coordinates of any segment.
- In order to find the midpoint, all one needs is to find the average of the x-coordinates and y- coordinates.
- The ratio on both sides of the midpoint is in the ratio of 1:1.
What are the expressions for finding the x and y coordinates of the midpoint of any segment?
An expression for the x-coordinate of the midpoint is (x1+x2)/2.
An expression for the y-coordinate of the midpoint is (y1+y2)/2.
Hence, the coordinates of the midpoint, (x,y) is given by [(x1+x2)/2, (y1+y2)/2].
Formulas in relation to midpoint
According to the centroid of the triangle formula, the centroid of the triangle is the same as the total of the x-coordinates of the vertices of the triangle divided by 3, and the total of the y-coordinates of the vertices of the triangle divided by 3, which is given as,
The centroid formula for triangle (x1+x2+x3)/3, (y1+y2+y3)/3
Derivation of the midpoint formula
Let the coordinates of endpoints of any segment be (x1, y1) and (x2, y2).
Then the middle point (at halfway) is (x1+x2)/2, (y1+y2)/2.
Thus, we derive the same.
Example:
Assume that a segment has endpoints (-8,1) and (4,5), then the midpoint of the segment lies on the line y=-x+1?
Solution: Now, we can find the midpoint of the coordinates by using the midpoint formula and that comes out to be [(-8+4)/2, (1+5)/2] which is (-2,3).
The coordinates (-2,3) satisfy the equation of the line thus proving the question.
Whenever one wants to find the middle point of two other points on the coordinate plane, ( for example if we want to find a line that divides the given line into two equal halves ) we can use the midpoint formula. The real-world problems find easy solutions by the application of distance and midpoint formulas.
As long as one remembers the midpoint formula as averaging the two points, all will be simple and good. In geometry, the middle point of a line segment is the midpoint. It is equidistant from the endpoints. The Cuemath website describes and explains the concept of the midpoint formula practically with examples and descriptions. The experts and teachers explain the concept in an efficient manner.
Note: It should be taken into consideration that which coordinate comes first doesn’t matter but it should be taken care that x is added to x and y to y.
Example: Calculating p for (–2, 2.5) is the midpoint between (p, 2) and (–1, 3).
Apply the Midpoint Formula:
{p + (-1)}/{2},{2 + 3}/2= (-2,2.5)
i.e., (p-1)/2=-2 and 5/2=2.5 which is already sorted
So, calculating for p we get p= -3.
The y-coordinates already match. This gives the value necessary for making the x-values match. So:
So the answer is p = –3. Answer
The concept is also useful in calculating the centre of the circle if the coordinates of the diameter are known.
For example: Calculate the coordinates of the centre of a circle whose endpoint coordinates of the diameter are (0,2), (3,4).
Sol: Given coordinates of the end points (0,2)& (3,4)
(x1,y1) = (0, 2)
(x2, y2)= (3, 4)
Coordinates of the centre of a circle= [(0+3)/2, (2+4)/2] as per midpoint formula
= [3/2, 6/2]
=[1.5,3] :Answer
Summing up, Midpoint Formula: The midpoint of the line segment whose endpoints are the two points (x1,y1) and (x2,y2) is. (x1+x22,y1+y22).