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1. Foundations of Geometry
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Geometry

1.04 Line segments and constructions

Lesson
Practice

Line segments

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 L M is marked with a dash and labeled 4 centimeters. Segment M N is also marked with a dash and labeled 4 centimeters.
Midpoint

Divides a segment into two congruent segments.

Segment A C with point B on A C. B is labeled Midpoint.
Segment bisector

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

Segment A B with midpoint M. Segment C D intersects A B at M.

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.
Worked Solution
Create a strategy

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

Apply the idea

XZ=3

Reflect and check

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

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle QS\displaystyle =\displaystyle QR+RSSegment addition postulate
\displaystyle QS\displaystyle =\displaystyle 33+30Substitute QR=33 and RS=30
\displaystyle QS\displaystyle =\displaystyle 63Evaluate the addition
b

PQ

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle PQ+QR\displaystyle =\displaystyle PRSegment addition postulate
\displaystyle PQ+33\displaystyle =\displaystyle 57Substitute QR=33 and PR=57
\displaystyle PQ\displaystyle =\displaystyle 24Subtract 33 from both sides

Example 3

Point B bisects \overline{AC}.

a

Identify two congruent segments.

Worked Solution
Create a strategy

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.
Apply the idea

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

b

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

Worked Solution
Create a strategy

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.

Apply the idea
\displaystyle AB+BC\displaystyle =\displaystyle ACSegment addition postulate
\displaystyle 7+7\displaystyle =\displaystyle ACSubstitute AB=7 and BC=7
\displaystyle 14\displaystyle =\displaystyle ACEvaluate the addition
Idea summary

Line segments can be measured using the ruler postulate by aligning the segment with a number line and finding the difference between the values each point lines up with. We can solve problems involving line segments using the segment addition postulate. Congruent segments can be labeled with congruency marks. Midpoints and bisectors are two ways a segment can be divided into congruent segments.

Geometric constructions

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 endpoints.
  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.

Examples

Example 4

Construct a copy of \overline{GH}.

Segment G H.
Worked Solution
Create a strategy

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

Apply the idea

Using technology:

A screenshot of the GeoGebra geometry tool showing segment G H and a point. Speak to your teacher for more details.

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

A screenshot of the GeoGebra geometry tool showing segment G H and its length and a point. Speak to your teacher for more details.

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

A screenshot of the GeoGebra geometry tool showing segment G H and its copy. Speak to your teacher for more details.

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

Reflect and check

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.

Rectangle A B C D.

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

Worked Solution
Create a strategy

A line that divides the sheep enclosure into two congruent rectangular sections must bisect 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 perpendicular bisector of \overline{AB} and extending it to reach \overline{CD}. We will do so in this case by making use of technology.

Apply the idea
A screenshot of the GeoGebra geometry tool showing rectangle A B C D. A circle centered at A is drawn. Speak to your teacher for more details.

We start by creating an arc centered at A that has a radius that 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.

A screenshot of the GeoGebra geometry tool showing rectangle A B C D. Two circles centered at A and B are drawn. Speak to your teacher for more details.

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.

A screenshot of the GeoGebra geometry tool showing rectangle A B C D. Two circles centered at A and B are drawn. The intersection of the two circles are drawn. Speak to your teacher for more details.

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

A screenshot of the GeoGebra geometry tool showing rectangle A B C D. Two circles centered at A and B are drawn. The intersection of the two circles are drawn. A line is drawn using the points of intersection. Speak to your teacher for more details.

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

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).

Reflect and check

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:

A screenshot of the GeoGebra geometry tool showing rectangle A B C D. Two circles centered at A and D are drawn. The intersection of the two circles are drawn. A line is drawn using the points of intersection. Speak to your teacher for more details.
Idea summary

Geometric figures can be constructed using a compass and straight edge. We can copy a line segment to any point using congruent circles and bisect a line segment using overlapping congruent circles.

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