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.

- The
**coordinate**is the number associated with the point. So if the coordinates of points A and B are a and b, then the distance between A and B is |a–b| or |b–a|.

The line segment on this ruler is 5 cm long, so* *\(AB = m\overline{AB} = 5 cm\)

- We don’t have to start at the zero mark on a ruler to measure the distance or length, but it is easier to measure distance by starting at the zero mark!

*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.

- Distance =

If two points line up vertically, the change in the y-coordinates will be the distance between the points.

- Distance =

The general distance formula to find the length between two points is:

- d = \(\sqrt{(\color{blue}{x_1 - x_2} \color{black}{)^2} + (\color{red}{y_1 - y_2}\color{black}{)^2}}\)

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.

- If \(AB = CD\), then \(\overline{AB} \cong \overline{CD}\)

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 \(AB = CD\)

- Do not assume a point is the midpoint if there are no tick marks showing the segments are congruent!

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** (**\( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\))

- The midpoint is the average of the x-values and the average of the y-values.

Example: