# Segments

## Big Picture

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.

## Key Terms

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.

## Distances

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

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|.

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 $$AB = m\overline{AB} = 5 cm$$

• 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.

# Segments cont.

## Distances on Grids

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 = |-4 - 3| = 7

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

• Distance = |9 - 3| = 6

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

## Congruent Segments

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

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: