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1. Foundations of Geometry
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Geometry

1.02 Segments

Lesson
Practice
Lesson

Concept summary

There are two postulates in Geometry that allow us to measure and solve problems involving segment lengths.

Ruler postulate

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.

Segment A C with point B on A 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.

A line segment with endpoints A and C with a point B directly in the middle labeled Midpoint
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.
Perpendicular bisector

A line, segment, or ray that is perpendicular to a segment at its midpoint.

A vertical line passing through a horizontal line segment. The line splits the segment into two smaller congruent segments. The angle where the line and the segment intersect has a right angle marking.

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:

  1. Identify the segment we want to copy.
  2. Draw a point that will become the first endpoint of the copied segment.
  3. Open the compass width to measure the distance between end points.
  4. 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.
  5. 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:

  1. Identify the segment we want to bisect.
  2. 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.
  3. Without changing the compass width, draw another arc from the other endpoint that intersects the original arc on both sides of the segment.
  4. Label the intersection of the arcs with points.
  5. Connect the points.
A diagram showing the 5 steps of constructing the bisector of a segment. Speak to your teacher for more information.

Note that this construction produces a bisector which is a perpendicular bisector.

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.
a

QS

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+RSSegment addition postulate
\displaystyle QS\displaystyle =\displaystyle 33+30Substitute known values
\displaystyle QS\displaystyle =\displaystyle 63Simplify
b

PQ

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

Example 3

Point B bisects \overline{AC}.

a

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}

b

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 ACSegment addition postulate
\displaystyle 7+7\displaystyle =\displaystyle ACSubstitute known values
\displaystyle 14\displaystyle =\displaystyle ACSimplify

Example 4

Construct a copy of segment \overline{GH}.

Segment G H

Approach

To construct a copy of a segment, we can follow the steps detailed in the concept summary. We can do it by hand or using technology in the form of GeoGebra.

Solution

Using technology:

First we use the point tool to add a new point, which will be one endpoint of the copied segment.

Next we can use the Distance or Length measurement tool to find the length of the original segment GH. This is the equivalent of measuring the segment with a compass.

Finally, we can use the Segment with Given Length tool to create a segment from our new point using the length that we measured.

Reflection

Note that there are many other tools in GeoGebra that we could use to construct a copy of a segment. For example, after using the measurement tool to find the length GH, we could then use the Circle: Center & Radius tool to create a circle with a radius of 4.9 units. Any radius of that circle would then be a copy of the original segment.

Example 5

Ursula has a rectangular fenced sheep enclosure, shown below. She wants to build an additional fence that will divide the enclosure into two congruent rectangular sections.

A reactangle A B C D.

Construct a line to represent the additional fence, showing all steps.

Approach

A line which divides the sheep enclosure into two congruent rectangular sections must biesct a pair of opposite sides of the enclosure. In particular, the line will be a perpendicular bisector of both sides.

So we can draw a line to divide the enclosure into two congruent rectangular sections by constructing a perpedicular bisector of \overline{AB} and extending it to reach \overline{CD}. We will do so in this case by making use of technology.

Solution

We start by creating an arc centered at A that has a radius which is greater than half the length of \overline{AB}.

The easiest way to achieve this is to use the Circle: Center & Radius tool and manually input the radius. This way we will be able to create an arc with an identical radius at the next step.

We now create an identical circle by using the Circle: Center & Radius tool and inputting the same radius, but this time centered at the other endpoint B.

Next we use the Point tool to mark both intersections of these two arcs.

Finally, we can draw a line that passes through these two points of intersection.

In order to divide the rectangle into two smaller congruent rectangles, we need to make sure this line extends all the way to intersect \overline{CD}, so we use the Line tool (rather than the Segment tool).

Reflection

We could also have divided the sheep enclosure into two congruent rectangles by bisecting the other sides, \overline{AD} and \overline{BC}. The result of this construction would look like the following:

Outcomes

G.CO.C.8

Use definitions and theorems about lines and angles to solve problems and to justify relationships in geometric figures.

G.CO.D.11

Perform formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

G.CO.D.12

Use geometric constructions to solve geometric problems in context, by hand and using technology.*

G.MP1

Make sense of problems and persevere in solving them.

G.MP2

Reason abstractly and quantitatively.

G.MP3

Construct viable arguments and critique the reasoning of others.

G.MP4

Model with mathematics.

G.MP5

Use appropriate tools strategically.

G.MP6

Attend to precision.

G.MP7

Look for and make use of structure.

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