Line Segments: Two Endpoints and a Measurable Length
A line segment is the part of a line that starts at one point and ends at another. Those two points are called the endpoints. Unlike a line (which extends forever), a segment has a finite, measurable length.
Defining the segment \(\overline{AB}\)
The segment with endpoints \(A\) and \(B\) is written as \(\overline{AB}\). In a diagram, it is drawn as a straight path from point \(A\) to point \(B\), with solid dots (or clear labeled points) at the ends to show the endpoints.
| What you mean | How you write it | What it represents |
|---|---|---|
| The segment itself | \(\overline{AB}\) | The set of points between and including \(A\) and \(B\) |
| The length (a number) | \(AB\) | The distance from \(A\) to \(B\), measured in units (cm, in, etc.) |
Segment vs. its length: \(\overline{AB}\) versus \(AB\)
It is important to separate the object from its measurement:
- \(\overline{AB}\) names the geometric figure (the segment).
- \(AB\) names the numerical value of its length (a distance).
Example: If a ruler shows that the distance from \(A\) to \(B\) is 6 cm, you would write \(AB = 6\text{ cm}\). You are not saying the segment equals 6 cm; you are saying its length equals 6 cm.
Congruent Segments and Tick Marks
Two segments are congruent if they have the same length. Congruence is about equality of lengths, even if the segments are in different locations or directions in a figure.
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How to state congruence
If segment \(\overline{AB}\) has the same length as segment \(\overline{CD}\), you can write:
\(\overline{AB} \cong \overline{CD}\)
This statement means the segments are congruent. It implies the lengths are equal:
\(AB = CD\)
How diagrams show congruent segments
Diagrams often use tick marks to show congruent segments:
- Segments with one tick mark are congruent to each other.
- Segments with two tick marks are congruent to each other (but not necessarily to the one-tick group).
- Different numbers of tick marks represent different congruence groups.
Example interpretation: If \(\overline{AB}\) and \(\overline{CD}\) each have one tick mark, then \(\overline{AB} \cong \overline{CD}\). If \(\overline{EF}\) has two tick marks, it is congruent to any other segment with two tick marks, not to the one-tick segments.
Midpoints: Splitting a Segment into Two Equal Parts
Definition and notation
A midpoint of a segment is the point on the segment that divides it into two congruent segments.
If \(M\) is the midpoint of \(\overline{AB}\), then:
- \(M\) lies on \(\overline{AB}\) (it is between \(A\) and \(B\)).
- \(\overline{AM} \cong \overline{MB}\).
- Equivalently, the lengths are equal: \(AM = MB\).
How to show a midpoint in a diagram
To show that \(M\) is the midpoint of \(\overline{AB}\) in a diagram, you typically:
- Place point \(M\) on the segment between \(A\) and \(B\).
- Add matching tick marks on \(\overline{AM}\) and \(\overline{MB}\) to show they are congruent.
Common statement format:
M is the midpoint of \(\overline{AB}\), so \(AM = MB\).
Practical Measurement Activities (Ruler: Real or Simulated)
Activity 1: Measuring a segment and recording its length
Goal: Measure segments accurately and write lengths with units.
- Draw or view a segment \(\overline{AB}\) on paper or on-screen.
- Align the ruler so the 0 mark is exactly at endpoint \(A\). If the segment does not start at 0 on a printed ruler, use subtraction (see step 5).
- Read the scale at endpoint \(B\). Decide whether you are measuring in centimeters or inches and stay consistent.
- Record the measurement as a length statement, for example:
\(AB = 7.4\text{ cm}\). - If the segment starts at a nonzero mark, read both endpoints and subtract. Example: if \(A\) is at 2.0 cm and \(B\) is at 9.1 cm, then \(AB = 9.1 - 2.0 = 7.1\text{ cm}\).
Activity 2: Comparing segments by measurement
Goal: Decide which segments are longer/shorter and which are equal in length.
- Measure \(\overline{AB}\), \(\overline{CD}\), and \(\overline{EF}\) using the same unit.
- Write each length with units:
\(AB = \_\_\_\),\(CD = \_\_\_\),\(EF = \_\_\_\). - Compare the numbers to decide relationships such as
\(AB > CD\),\(EF = CD\), etc. - If two lengths match (within reasonable measuring accuracy), state congruence:
\(\overline{AB} \cong \overline{CD}\).
Activity 3: Finding a midpoint by measurement
Goal: Locate a midpoint using distance.
- Measure the full length of \(\overline{AB}\). Example:
\(AB = 10\text{ cm}\). - Compute half the length:
\(\frac{AB}{2} = 5\text{ cm}\). - Starting at \(A\), mark a point \(M\) that is 5 cm from \(A\) along the segment.
- Check by measuring \(MB\). It should also be 5 cm (or extremely close):
\(AM = MB\). - State the result:
M is the midpoint of \(\overline{AB}\), so \(AM = MB\).
Practice Problems
1) Identify segments in a figure
In the figure, points \(A\), \(B\), \(C\), and \(D\) lie on the same straight path in that order. List four different segments that can be named using these points.
- Write your answers using segment notation (with a bar), such as \(\overline{AB}\).
2) Segment vs. length statements
Decide whether each statement is written correctly. If not, rewrite it correctly.
- a) \(\overline{AB} = 8\text{ cm}\)
- b) \(AB = 8\text{ cm}\)
- c) \(\overline{AB} \cong \overline{CD}\) and \(AB = CD\)
3) Congruent segments from tick marks
A diagram shows:
- \(\overline{PQ}\) and \(\overline{RS}\) each have one tick mark.
- \(\overline{TU}\) and \(\overline{VW}\) each have two tick marks.
- \(\overline{XY}\) has no tick marks.
Answer:
- a) Write a congruence statement for the one-tick segments.
- b) Write a congruence statement for the two-tick segments.
- c) Can you conclude anything about \(\overline{XY}\) compared to the others? Explain using the markings idea.
4) Midpoint reasoning
Point \(M\) lies on \(\overline{AB}\). For each prompt, write the correct statement(s).
- a) If \(M\) is the midpoint of \(\overline{AB}\), write an equation involving lengths.
- b) If \(AM = MB\) and \(M\) lies on \(\overline{AB}\), what can you conclude about \(M\)?
5) Mixed measurement and congruence
You measure three segments and record:
- \(JK = 5.2\text{ cm}\)
- \(LM = 5.2\text{ cm}\)
- \(NO = 6.0\text{ cm}\)
Answer:
- a) Which segments are congruent? Write a congruence statement.
- b) Which segment is longest? Write an inequality using lengths.
- c) If \(P\) is the midpoint of \(\overline{NO}\), what is \(NP\)? Include units.
