Triangle Proportionality Theorem Calculator
Calculate missing segments using the triangle proportionality theorem (side-splitter theorem)
Triangle Proportionality Calculator
Triangle Setup
Consider triangle ABC with line DE parallel to side BC, where:
AD: Segment from A to D on side AB
DB: Segment from D to B on side AB
AE: Segment from A to E on side AC
EC: Segment from E to C on side AC
Proportionality Theorem: AD/DB = AE/EC
Distance from vertex A to point D
Distance from vertex A to point E
Distance from point D to vertex B
Distance from point E to vertex C
Example Calculation
Find Missing Segment
Given: AD = 6, AE = 8, DB = 9
Find: EC
Solution:
Using Triangle Proportionality Theorem: AD/DB = AE/EC
6/9 = 8/EC
EC = (8 × 9) / 6 = 72 / 6 = 12
Answer: EC = 12
Verification
Check proportionality:
AD/DB = 6/9 = 0.6667
AE/EC = 8/12 = 0.6667
Result: Since the ratios are equal, DE is parallel to BC
Triangle Proportionality Theorem
Statement
If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally.
Formula
AD/DB = AE/EC
Converse
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
Key Concepts
Side-Splitter Theorem
Another name for the triangle proportionality theorem
Parallel Lines
The intersecting line must be parallel to the base
Proportional Segments
The ratios of corresponding segments are equal
Similar Triangles
Creates similar triangles with proportional sides
Understanding the Triangle Proportionality Theorem
What is the Triangle Proportionality Theorem?
The Triangle Proportionality Theorem, also known as the Side-Splitter Theorem, states that if a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those two sides proportionally.
Applications
- •Finding missing segment lengths in triangles
- •Proving lines are parallel to triangle sides
- •Solving problems involving similar triangles
- •Engineering and architectural measurements
Mathematical Proof
Given: Triangle ABC with DE parallel to BC
Prove: AD/DB = AE/EC
1. ∠ADE ≅ ∠ABC (corresponding angles)
2. ∠AED ≅ ∠ACB (corresponding angles)
3. △ADE ~ △ABC (AA similarity)
4. AD/AB = AE/AC (corresponding sides)
5. AD/(AD+DB) = AE/(AE+EC)
6. Cross multiply and simplify to get AD/DB = AE/EC
Note: This theorem is fundamental in proving many other geometric relationships and is widely used in coordinate geometry and trigonometry.