The Midpoint Formula
The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. It divides the segment into two equal halves. The midpoint formula averages the x-coordinates and y-coordinates separately.
Midpoint M = ( (xโ + xโ) / 2 , (yโ + yโ) / 2 )
Example: A(2,4) and B(8,10) โ M = ((2+8)/2, (4+10)/2) = (5, 7)
Example: A(2,4) and B(8,10) โ M = ((2+8)/2, (4+10)/2) = (5, 7)
Worked Examples
Example 1 โ Basic Midpoint
Find the midpoint between A(1, 3) and B(7, 9).
M = ((1+7)/2, (3+9)/2) = (8/2, 12/2) = (4, 6)
Example 2 โ Negative Coordinates
Find the midpoint between P(โ4, 6) and Q(10, โ2).
M = ((โ4+10)/2, (6+(โ2))/2) = (6/2, 4/2) = (3, 2)
Example 3 โ Find an Endpoint from Midpoint
The midpoint of AB is M(5, 3). If A is at (2, 1), what is B?
5 = (2 + xโ)/2 โ xโ = 10 โ 2 = 8
3 = (1 + yโ)/2 โ yโ = 6 โ 1 = 5
Therefore B = (8, 5)
3 = (1 + yโ)/2 โ yโ = 6 โ 1 = 5
Therefore B = (8, 5)
Example 4 โ Midpoint of a Room Diagonal
A rectangular room has corners at (0,0) and (6,4). Where is the center of the room?
Center = ((0+6)/2, (0+4)/2) = (3, 2)
Properties of the Midpoint
- The midpoint always lies on the line segment between the two points
- The distance from each endpoint to the midpoint is equal (half the total distance)
- The midpoint of a diameter of a circle is the center of the circle
- In a triangle, the midpoints of the three sides connected form the medial triangle (with 1/4 the area)
Distance from A to Midpoint = Distance from Midpoint to B = (1/2) ร AB
The Midpoint Theorem in Triangles
The Triangle Midpoint Theorem states: the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and exactly half its length. This is widely used in proofs and geometry constructions.
Real-World Applications
- Construction: Finding the center of walls, floors, windows for symmetrical placement
- Navigation: Finding halfway point between two geographic locations
- Computer graphics: Bรฉzier curves, line bisection, rendering midpoints
- Sports: Finding the center spot of a football pitch or basketball court
- Logistics: Finding the optimal meeting point between two locations