Every point on a line can be paired with a real number. This makes a one-to-one correspondence between the points on the line and the real numbers. The real number that corresponds to a point is called the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference of the coordinates of A and B.

Segment A B on a number line. Point A is plotted at x, and B is plotted at y. The equation A B equals the absolute value of x minus y is shown.

Segment addition postulate

If B is between A and C, then AB + BC = AC.

If AB + BC = AC, then B is between A and C.

A segment can be bisected, which means it has been divided into two congruent segments. The midpoint, which bisects a segment, is the point exactly halfway between the two endpoints of a segment.

Congruent segments

Two segments whose measures are equal. Labeled with congruent marks.

A line segment with endpoints L and N with a point M directly in the middle. Segment LM is marked with a dash and labeled 4 centimeters. Segment MN is also marked with a dash and labeled 4 centimeters

Midpoint

A point exactly halfway between the endpoints of a segment that divides it into two congruent line segments.

Segment bisector

A line, segment, ray, or plane that intersects a segment at its midpoint.

A line segment with endpoints A and B with a point M directly in the middle. Segment AM is marked with one dash and segmnt MB is marked with one dash. There is a line that passes through point M.

A geometric construction is the accurate drawing of angles, lines and shapes. The tools used for these constructions are a straightedge, compass and pencil.

To construct a copy of a segment, we will:

Identify the segment we want to copy.
Draw a point that will become the first endpoint of the copied segment.
Open the compass width to measure the distance between end points.
Without changing the compass width, place the point end of the compass on the point we constructed for the copy and draw a small arc. Place a point anywhere on the arc.
Connect the two points using a straightedge.

A diagram showing the 5 steps of constructing a copy of a segment. Speak to your teacher for more information.

To construct the bisector of a segment, we will:

Identify the segment we want to bisect.
Open the compass width to just past half the segment's length and draw an arc from one endpoint that extends to both sides of the segment.
Without changing the compass width, draw another arc from the other endpoint that intersects the original arc on both sides of the segment.
Label the intersection of the arcs with points.
Connect the points.

A diagram showing the 5 steps of constructing the bisector of a segment. Speak to your teacher for more information.

Worked examples

Example 1

Use the ruler postulate to find the length of \overline{XZ}.

A number line ranging from 0 to 8 in steps of 1. A point labeled X is on the 2 mark, a point labeled Y on the 4 mark, and a point labeled Z on the 5 mark.

Approach

Since X lines up with the real number 2 and Z lines up with the real number 5, the length of segment \overline{XZ} is the difference between 2 and 5.

Solution

XZ=3

Reflection

The length of a line segment will not include the line segment symbol: XZ=3 but not \overline{XZ}=3.

Example 2

Use the segment addition postulate to find the length of the following:

Segment P S with points Q and R on P S. Q is between P and R, and R between Q and S. P R has a length of 57, Q R has a length of 33, and R S has a length of 30.

Approach

We know that QR=33 and RS=30. Since R is between Q and S, by the segment addition postulate we know that QS=QR+RS.

Solution

\displaystyle QS	\displaystyle =	\displaystyle QR+RS	Segment addition postulate
\displaystyle QS	\displaystyle =	\displaystyle 33+30	Substitute known values
\displaystyle QS	\displaystyle =	\displaystyle 66	Simplify

Approach

We know that PR=57 and QR=33. Since Q is between P and R, by the segment addition postulate we know that PQ+QR=PR

Solution

\displaystyle PQ+QR	\displaystyle =	\displaystyle PR	Segment addition postulate
\displaystyle PQ+33	\displaystyle =	\displaystyle 57	Substitute known values
\displaystyle PQ	\displaystyle =	\displaystyle 24	Subtract 33 from both sides

Example 3

Point B bisects \overline{AC}.

Identify two congruent segments.

Approach

To bisect something is to divide it into two congruent parts. Since B bisects \overline{AC}, it creates two congruent segments as shown in the diagram.

Segment A C with point B on A C. A B and B C have the same lengths.

Solution

\overline{AB}\cong \overline{BC}

If AB=7, find the length of \overline{AC}.

Approach

Using the segment addition postulate we know that AB+BC=AC. Since B bisects \overline{AC} we know that \overline{AB}\cong \overline{BC} which tells us that BC=7.

Solution

\displaystyle AB+BC	\displaystyle =	\displaystyle AC	Segment addition postulate
\displaystyle 7+7	\displaystyle =	\displaystyle AC	Substitute known values
\displaystyle 14	\displaystyle =	\displaystyle AC	Simplify

Outcomes

MA.912.GR.1.1

Prove relationships and theorems about lines and angles. Solve mathematical and real-world problems involving postulates, relationships and theorems of lines and angles.

MA.912.GR.5.1

Construct a copy of a segment or an angle.

MA.912.GR.5.2

Construct the bisector of a segment or an angle, including the perpendicular bisector of a line segment.

1.02 Measuring segments

Concept summary

Worked examples

Example 1

Approach

Solution

Reflection

Example 2

Approach

Solution

Approach

Solution

Example 3

Approach

Solution

Approach

Solution

Outcomes

MA.912.GR.1.1

MA.912.GR.5.1

MA.912.GR.5.2

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