The Segment Addition Postulate
Master the Fundamental Building Block of Geometry
📐 What is the Segment Addition Postulate?
The Segment Addition Postulate is a fundamental principle in geometry that states: if three points A, B, and C are collinear (lie on the same line) such that point B lies between points A and C, then the sum of the lengths of segments AB and BC equals the length of the entire segment AC.
In simpler terms: If you divide a line segment into smaller parts, the sum of those parts equals the whole segment.
⭐ The Main Formula
AB + BC = AC
Where B lies between A and C on the line segment
👁️ Visual Understanding
Point B divides segment AC into two parts: AB and BC
🔢 Extended Formula (Multiple Points)
When there are multiple points between the endpoints, extend the postulate:
AB + BC + CD = AD
AB + BC + CD + DE = AE
✨ Key Properties & Facts
1️⃣ Collinearity Required
All points must lie on the same line for the postulate to apply.
2️⃣ No Proof Needed
It's a postulate (axiom), meaning it's accepted as true without proof.
3️⃣ Line Segments Only
Applies only to line segments, not rays or infinite lines.
4️⃣ Reversible
If AB + BC = AC, then B must lie between A and C.
🎯 Common Applications
🔍 Finding Missing Lengths
Use: AC - AB = BC or AC - BC = AB
🔍 Identifying Midpoints
If B is the midpoint of AC, then AB = BC and AC = 2AB = 2BC
🔍 Solving Algebraic Equations
Set up equations when segments are expressed with variables
💡 Math Tricks & Problem-Solving Tips
✅ Trick #1: Draw It Out
Always sketch the line segment with labeled points. Visual representation makes solving 10x easier!
✅ Trick #2: Write the Formula First
Before substituting numbers, write: AB + BC = AC. Then plug in what you know.
✅ Trick #3: Isolate the Unknown
Rearrange to solve: Unknown = Total - Known part
✅ Trick #4: Check Collinearity
If AB + BC = AC, points are collinear. If not, they don't lie on the same line!
✅ Trick #5: Midpoint Shortcut
For midpoints, double the half-segment: AC = 2 × AB (when B is the midpoint)
📝 Step-by-Step Examples
Example 1: Finding a Missing Length
Problem: Point B lies on segment AC. If AB = 13 and BC = 8, find AC.
Step 1: Write the formula
AB + BC = AC
Step 2: Substitute the values
13 + 8 = AC
Step 3: Solve
AC = 21 units
✓ Answer: AC = 21 units
Example 2: Finding a Part When Total is Known
Problem: Point Q lies on segment PR. If PR = 45 and PQ = 28, find QR.
Step 1: Write the formula
PQ + QR = PR
Step 2: Substitute the values
28 + QR = 45
Step 3: Solve for QR
QR = 45 - 28 = 17
✓ Answer: QR = 17 units
Example 3: Solving with Algebraic Expressions
Problem: Point B lies on segment AC. If AB = 3x, BC = 2x + 4, and AC = 24, find x.
Step 1: Write the formula
AB + BC = AC
Step 2: Substitute the expressions
3x + (2x + 4) = 24
Step 3: Combine like terms
5x + 4 = 24
Step 4: Solve for x
5x = 20
x = 4
✓ Answer: x = 4
Example 4: Working with Midpoints
Problem: M is the midpoint of segment XY. If XM = 3x - 2 and MY = 2x + 3, find XY.
Step 1: Since M is the midpoint, XM = MY
3x - 2 = 2x + 3
Step 2: Solve for x
3x - 2x = 3 + 2
x = 5
Step 3: Find XM (or MY)
XM = 3(5) - 2 = 13
Step 4: Use Segment Addition Postulate
XY = XM + MY = 13 + 13 = 26
✓ Answer: XY = 26 units
⚠️ Important Notes to Remember
📌 Note 1: Order Matters
Point B must be between A and C, not outside the segment.
📌 Note 2: Units Consistency
Always ensure all measurements use the same units before applying the formula.
📌 Note 3: Check Your Answer
Verify that the sum of parts equals the whole. If not, recheck your calculations!
🎓 Interactive Quiz
Test your understanding with 10 practice questions!