Geometry

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|.
Ruler Postulate

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

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 \(\overline{AC} = \overline{A\color{red}{B}} + \overline{\color{red}{B} \color{black}{C}}\)

Segment Addition Postulate

If the points are not on a straight line, the Segment Addition Postulate does not apply.

Geometry

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
distances for horizontal lines

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

  • Distance = |9 - 3| = 6
distances for vertical lines

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!

 length between two points

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 \(\overline{AB} \cong \overline{CD}\)
Simplifying Rational Expressions

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  \(AB = CD\)

Ruler Postulate
  • 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

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:

 average of the x-values