Measuring Segments and Segment Addition
Quick Answer
The length of a segment can be found by measuring it or by finding the distance between the coordinates of its endpoints. If a point lies between two other points, the two smaller segment lengths add to equal the length of the entire segment.
Why this matters: Segment measurement is used throughout geometry to calculate distances, prove figures are congruent, locate midpoints, and solve real-world measurement problems.
Segment measurement is used throughout geometry to calculate distances, prove figures are congruent, locate midpoints, and solve real-world measurement problems.
Essential Question
How can segment lengths and the Segment Addition Postulate be used to find unknown distances?
Learning Goals
- 1Measure a segment accurately using a ruler or scale.
- 2Distinguish between a segment and its numerical length.
- 3Use coordinates to determine the distance between two points on a number line.
- 4Identify when a point lies between two other points.
- 5Apply the Segment Addition Postulate.
- 6Write and solve equations involving segment lengths.
- 7Check whether an answer is reasonable using the diagram and given information.
Key Vocabulary
Length of a segment
The distance between the two endpoints of a segment.
Distance
The nonnegative measure of how far apart two points are.
Coordinate
A number assigned to a point on a number line.
Ruler Postulate
Points on a line can be matched one-to-one with real numbers so that the distance between two points is the absolute value of the difference of their coordinates.
Between
Point B is between points A and C when A, B, and C are collinear and AB + BC = AC.
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
Congruent segments
Segments that have equal lengths.
Measurement
A numerical value describing the size or length of a geometric figure.
Units
The standard quantities used to express a measurement, such as inches, centimeters, or units.
Precision
The level of detail or smallest unit used in a measurement.
Key Ideas
Measuring a segment means finding the numerical distance between its two endpoints. This lesson connects geometric figures to numbers — a critical skill used throughout all of geometry.
A. Segment Versus Segment Length
- AB̄ (with an overbar) represents the geometric segment — the figure itself.
- AB (no overbar) represents the numerical length of the segment — a number.
- A segment is a figure; its length is a number with units.
- Example: AB̄ is a segment; AB = 5 cm is its length.
B. Measuring a Segment
- Place the ruler's zero mark at one endpoint.
- Read the location of the other endpoint.
- Include units in your answer (cm, in, units).
- Measure to the precision shown by the ruler.
- Do not assume a printed diagram is drawn to scale unless stated.
C. Coordinates and Distance
- On a number line, each point is assigned a real number called its coordinate.
- The distance between points A and B is: AB = |b − a|, where a and b are their coordinates.
- Distance is always nonnegative because absolute value is used.
- It does not matter which coordinate you subtract first — the absolute value makes the result the same.
D. The Ruler Postulate
- Every point on a line can be assigned a unique real-number coordinate.
- The distance between any two points equals the absolute value of the difference of their coordinates.
- This postulate is the formal justification for using a ruler to measure segments.
E. Points Between Other Points
- B is between A and C only when all three points are collinear and B lies on segment AC.
- The order may be written A–B–C or C–B–A — both describe the same arrangement.
- A diagram alone is not sufficient unless the labels and conditions support the relationship.
F. Segment Addition Postulate
- If B is between A and C, then: AB + BC = AC
- The two smaller parts equal the whole segment.
- This postulate works in both directions: if AB + BC = AC, then B is between A and C.
G. Solving for Unknown Lengths
- Identify the whole segment.
- Identify the two parts.
- Write the segment-addition equation: part + part = whole.
- Substitute known expressions.
- Solve the equation for the variable.
- Substitute back to find the requested segment lengths.
- Check: all lengths must be positive, and the parts must add to the whole.
H. Congruent Segments and Tick Marks
- Matching tick marks on a diagram indicate congruent segments.
- Congruent segments have equal numerical lengths.
- Do not assume two segments are congruent because they look the same length — only tick marks or given information confirm congruence.
Worked Examples
Problem: Segment AB is placed on a ruler with endpoint A at the zero mark. Endpoint B aligns with the 6 cm mark. What is AB?
Step 1: Confirm that A is at the zero mark. ✓
Step 2: Read the ruler at endpoint B. B aligns with 6 cm.
Step 3: Include units.
AB = 6 cm
Check: The measurement is positive and includes units. ✓
Problem: Point A has coordinate −3 and point B has coordinate 5. Find AB.
Step 1: Write the distance formula: AB = |b − a|
Step 2: Substitute: AB = |5 − (−3)|
Step 3: Simplify inside the absolute value: |5 + 3| = |8|
Step 4: Apply absolute value: |8| = 8
AB = 8 units
Check: Distance is nonnegative. ✓
Problem: B is between A and C. AB = 7 and BC = 5. Find AC.
Step 1: Write the Segment Addition Postulate: AB + BC = AC
Step 2: Substitute: 7 + 5 = AC
Step 3: Add: AC = 12
AC = 12
Check: 7 + 5 = 12 ✓
Problem: B is between A and C. AC = 18 and AB = 11. Find BC.
Step 1: Write the Segment Addition Postulate: AB + BC = AC
Step 2: Substitute: 11 + BC = 18
Step 3: Subtract 11 from both sides: BC = 18 − 11
BC = 7
Check: 11 + 7 = 18 ✓
Problem: B is between A and C. AB = 2x + 3, BC = x + 4, and AC = 19. Find x, AB, and BC.
Step 1: Write the Segment Addition Postulate: AB + BC = AC
Step 2: Substitute: (2x + 3) + (x + 4) = 19
Step 3: Combine like terms: 3x + 7 = 19
Step 4: Subtract 7: 3x = 12
Step 5: Divide by 3: x = 4
Step 6: Find AB: 2(4) + 3 = 11
Step 7: Find BC: 4 + 4 = 8
x = 4, AB = 11, BC = 8
Check: 11 + 8 = 19 ✓ Both lengths are positive. ✓
Multiple Choice Check
Choose the best answer for each question. Answers are in the Answer Key tab.
Which notation represents the length of a segment with endpoints A and B?
- AAB̄ (with an overbar)
- BAB (no overbar)
- C⃗AB (with an arrow)
- DAB↔ (with a double arrow)
Use the diagram below. Segment CD is placed on a ruler with C at the zero mark and D at the 9 cm mark. What is CD?
- A0 cm
- B9 cm
- C18 cm
- D4.5 cm
Point P has coordinate −4 and point Q has coordinate 6. What is PQ?
- A2
- B−10
- C10
- D−2
B is between A and C. AB = 8 and BC = 5. Which equation correctly represents the Segment Addition Postulate?
- AAB − BC = AC
- BAB + BC = AC
- CAB × BC = AC
- DAC + BC = AB
The diagram shows two segments with matching tick marks. What can you conclude?
- AThe segments are parallel.
- BThe segments are perpendicular.
- CThe segments are congruent — they have equal lengths.
- DThe segments are part of the same line.
Guided Practice
Work through each problem. Reveal the solution when you are ready to check.
Segment EF is placed on a ruler with E at the zero mark and F at the 8 cm mark. What is EF?
Point M has coordinate 2 and point N has coordinate 9. Find MN.
B is between A and C. AB = 6 and BC = 9. Find AC.
B is between A and C. AC = 20 and BC = 7. Find AB.
B is between A and C. AB = 3x + 1, BC = 2x − 1, and AC = 25. Find x, AB, and BC.
Common Mistakes
Confusing AB̄ (overbar) with AB (no overbar)
The overbar notation represents the geometric segment — the figure. AB without an overbar represents the numerical length — a number.
Subtracting coordinates without using absolute value
Distance is always nonnegative. Use AB = |b − a| so the result is never negative regardless of which coordinate is larger.
Using the Segment Addition Postulate when the points are not collinear
The postulate requires that B lies between A and C on the same line. Three non-collinear points do not satisfy this condition.
Adding the whole segment to one part instead of the two parts together
The two smaller parts add to equal the whole: AB + BC = AC. The whole is never added to a part.
Assuming the diagram is drawn to scale
Use the labeled measurements and given information, not the visual appearance of the diagram.
Stopping after finding x
If the problem asks for a segment length, substitute x back into the expression to find the actual length.
Forgetting units in the final answer
When units are given (cm, in, units), include them in the final measurement.
Before You Finish
Make sure you can answer yes to each of these:
- ✓Segment notation (overbar) and segment length (no overbar) are different things.
- ✓Distance between coordinates uses absolute value: AB = |b − a|.
- ✓Segment lengths are always nonnegative.
- ✓If B is between A and C, then AB + BC = AC.
- ✓The two parts equal the whole — never add the whole to a part.
- ✓Algebraic problems require solving for x and then finding the requested length.
- ✓Diagrams may not be drawn to scale — use labeled values.
Check Your Understanding
1. How is AB̄ different from AB?
2. If B is between A and C, what equation relates AB, BC, and AC?
3. Why is absolute value used when finding distance on a number line?