Line Segments and Midpoints

• A line segment is a piece of a line that starts and ends at specific points, called endpoints
• A line continues infinitely in both directions
• The midpoint of a line segment is the point exactly in the middle, dividing it into two equal parts
• Lines do not have midpoints

Vertical Line Segments

• For a vertical line segment with endpoints (2,1) and (2,7), the midpoint's x-coordinate is 2
• The y-coordinate is the average of 1 and 7, which is 4
• The midpoint of the segment is (2,4)

Horizontal Line Segments

• For horizontal and vertical lines, one coordinate remains the same
• The other coordinate of the midpoint is the average of the corresponding coordinates of the endpoints

Diagonal Line Segments

• The midpoint of a diagonal line is found by averaging both sets of coordinates of the endpoints
• For a line segment from (3,0) to (5,6), the x-coordinate of the midpoint is the average of 3 and 5, which is 4
• The y-coordinate of the midpoint is the average of 0 and 6, which is 3
• The midpoint of the line segment with endpoints (3,0) and (5,6) is (4,3)

Midpoint Formula

• The midpoint of a line segment is given by the formula:

, where and are the endpoints

• The order of the endpoints in the formula does not affect the result

Example 1

• To find the midpoint of a line segment with endpoints (-4,3) and (1,-3), use the midpoint formula
• Midpoint =

= (-1.5, 0)

Example 2

• To find the midpoint of a line segment with endpoints (1.6,-5) and (-6,-2.2), apply the midpoint formula
• Midpoint =

= (-2.2, -3.6)

Finding an Endpoint

• If given a midpoint and one endpoint, the midpoint formula can be used to find the missing endpoint
• Given endpoint (-4,2) and midpoint (-1,1), set up the formula:

= (-1,1)

• Split into two equations:

  and

• Solving these equations gives x = 2 and y = 0, so the other endpoint is (2,0)