A segment is a section of a line between two points. Segments have many properties, but the most important property to understand is that line segments have a start point and an end point. Line segments can be bisected by other segments or lines. Often times, we will be asked to find the length of a segment. Two segments can be added together so that the length of the new segment is the sum of the lengths of the old segments.
Segment: Portion of a line that is ended by two points.
Endpoint: Point at the end of a segment or at the start of a ray.
Distance: How far apart two geometric objects are.
Coordinate: The real number that corresponds to a point.
Congruent Segments: Two segments that have the same length.
Midpoint: A point on a line segment that divides the segment into two congruent segments.
Segment Bisector: A line, segment, or ray that passes through a midpoint of another segment.
The distance is the length between two points. Distance is ALWAYS positive!
A line segment is labeled \(\overline{AB}\), but the distance of the line segment (the distance between endpoints A and B) is labeled as AB or \(m\overline{AB}\). The m stands for “measure.”
Ruler Postulate: Using a ruler to find the distance between two points, the distance will be the absolute value of the difference between the numbers shown on the ruler.
Warning: Do not use the edge of the ruler or any portion of the ruler left of the zero mark in your measurements!
The line segment on this ruler is 5 cm long, so \(AB = m\overline{AB} = 5 cm\)
Segment Addition Postulate: The measure of any line segment can be found by adding the measures of the smaller segments that make it up.
If B is between A and C, then \(\overline{AC} = \overline{A\color{red}{B}} + \overline{\color{red}{B} \color{black}{C}}\)
If the points are not on a straight line, the Segment Addition Postulate does not apply.
Using the Ruler Postulate, we can find the distances for horizontal and vertical lines plotted in the x - y plane.
If two points line up horizontally, the change in the x-coordinates will be the distance between the points.
If two points line up vertically, the change in the y-coordinates will be the distance between the points.
The general distance formula to find the length between two points is:
Be careful of signs when working with negative coordinates!
See Geometry Study Guide: Pythagorean Theorem for a proof of this formula
When two segments have the same length, they are congruent. The symbol \(\cong\) is used to show congruence
To label congruent segments in a diagram, tick marks are used. Segments that are congruent will have the same number of tick marks.
Use = to mean equality for numbers and \(\cong\) to mean congruence for figures.
Midpoint Postulate: Any line segment will have exactly one midpoint.
B is the midpoint of \(\overline{AC}\) if \(AB = CD\)
A segment bisector is any ray, segment, or line that intersects the segment at its midpoint.
Midpoint formula: The midpoint for two points \((x_1, y_1)\) and \((x_2, y_2)\) is (\( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\))
Example: